ALBERT

Beschreibung: 〈p〉Publication date: Available online 27 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Romain Fiévet, Hughes Deniau, Estelle Piot〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Characteristic boundary conditions for the Navier-Stokes equations (NSCBC) are implemented for the first time with discontinuous spectral methods, namely the spectral difference and flux reconstruction. The implementation makes use of the resolution by these methods of the strong form of the Navier-Stokes equations by applying these conditions through a flux balance regularization which takes the form of a generalized element-compact correction polynomial. It is shown to be at least as effective as similar implementations in finite volume solvers, and sustains arbitrarily-high orders of accuracy on hexahedral-based unstructured meshes. Further, Navier-Stokes time-domain impedance boundary conditions are derived and implemented as a NSCBC sub-class. They account for the diffusive process at the wall and are shown to properly resolve broadband impedance models under normal and grazing flow conditions. The ability of these NSCBC in preventing the appearance of spurious reflections at the boundaries is demonstrated through a varied series of bench-marking simulations. They effectively shield the inner computational domain from any far-field unphysical contamination. Overall, this work enables the use of strong discontinuous spectral methods to study unsteady problems on complex geometries.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: Available online 27 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Lotte Romijn, Jan ten Thije Boonkkamp, Wilbert IJzerman〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We present a method for the design of a single freeform reflector that converts the light distribution of a point source to a desired light distribution in the far field. Using the geometrical-optics law of reflection and requiring energy conservation, this optical design problem can be represented by a generalized Monge-Ampère equation for the shape of the reflector with transport boundary condition. We use a generalized least-squares algorithm that can handle a logarithmic cost function in the corresponding optimal transport problem. The algorithm first computes the optical map and subsequently constructs the optical surface. We demonstrate that the algorithm can generate reflector surfaces for a number of complicated target distributions.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: Available online 24 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Fujun Liu, Haitao Dong〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper constructs a second-order large time step wave adding scheme (LTS-WA2) for hyperbolic conservation laws. Based on the first-order large time step wave adding scheme (LTS-WA1), a piece-wise linear reconstruction with limiter is performed on the solutions, and the band decomposition and band adding is complemented (into the discontinuity decomposition and wave adding), then the scheme is extended to second-order. The manufacture of the new scheme is simplified and written uniformly. Theoretical analyses are made for scalar cases. Numerical experiments give 11 tests which involve 1D scalar equations, 1D, 2D and 3D Euler equations. Computations show that the new scheme maintains high resolution and low dissipation of the original scheme at large CFL number, and also improves the problem of low resolution of the original scheme for rarefaction wave and contact discontinuity at small CFL number (roughly less than 2). Accuracy order is analyzed theoretically and validated using tests of 1D linear equation and 2D Burgers equation. CPU time is also compared with traditional second-order (Harten-TVD) scheme, showing that LTS-WA2 is more cost effective.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: Available online 24 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Fatemeh Ghasemi, Jan Nordström〈/p〉

Beschreibung: 〈p〉Publication date: Available online 24 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Weiming Li, Chang Liu, Yajun Zhu, Jiwei Zhang, Kun Xu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we extend the unified gas-kinetic wave-particle (UGKWP) method to multiscale photon transport. In this method, the photon free streaming and scattering processes are treated in an un-splitting way. The photon distribution is described by both discrete simulation particle and analytic distribution function. By accurately recovering the multiscale modeling of the unified gas-kinetic scheme (UGKS), the UGKWP method presents a smooth transition for photon transport from optically thin and optically thick regimes according to the cell's Knudsen number. In the optically thin regime, the UGKWP method performs as a Monte Carlo type particle tracking method, while in the optically thick regime it recovers a diffusion process without particles. The proportion of wave-described and particle-described photon is automatically adapted according to the numerical resolution and local photon scattering physics, i.e., the so-called cell Knudsen number. Compared to the discrete ordinates-based UGKS, the UGKWP method requires less memory and does not suffer from ray effect. Compared to the implicit Monte Carlo (IMC) method, the statistical noise of the UGKWP method is greatly reduced, and the computational efficiency is significantly improved in the optically thick regime. Several numerical examples covering all transport regimes from the optically thin to optically thick ones are computed to validate the accuracy and efficiency of the UGKWP method. In comparison with the UGKS and IMC method, the UGKWP method may have a several-order-of-magnitude reduction in computational cost and memory requirement in solving some multsicale transport problems.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: Available online 23 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Georgia Nykteri, Phoevos Koukouvinis, Silvestre Roberto Gonzalez Avila, Claus-Dieter Ohl, Manolis Gavaises〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A numerical methodology resolving flow complexities arising from the coexistence of both multi-scale processes and flow regimes is presented. The methodology employs the compressible Navier-Stokes equations of two interpenetrating fluid media using the two-fluid formulation; this allows for compressibility and slip velocity effects to be considered. On-the-fly criteria switching between a sharp and a diffuse interface within the Eulerian-Eulerian framework along with dynamic interface sharpening is developed, based on an advanced local flow topology detection algorithm. The sharp interface regimes with dimensions larger than the grid size are resolved using the VOF method. For the dispersed flow regime, the methodology incorporates an additional transport equation for the surface-mass fraction (Σ-ϒ) for estimating the interface surface area between the two phases. To depict the advantages of the proposed multiscale two-fluid approach, a high-speed water droplet impact case has been examined and evaluated against new experimental data; these refer to a millimetre size droplet impacting a solid dry smooth surface at velocity as high as 150 m/s, which corresponds to a Weber number of ∼7.6×10〈sup〉5〈/sup〉. Droplet splashing is followed by the formation of highly dispersed secondary cloud of droplets, with sizes ranging from 10〈sup〉−5〈/sup〉 mm close to the wall to less than 1 μm forming at the later stages of droplet fragmentation. Additionally, under the investigated impact conditions, compressibility effects dominate the early stages of droplet splashing. A strong shock wave forms and propagates inside the droplet, where transonic Mach numbers occur; local Mach numbers up to 2.5 are observed for the expelled surrounding gas outside the droplet. Relative velocities between the two fluids are also significant; local values on the tip of the injected water film up to 5 times higher than the initial impact velocity are observed. The proposed numerical approach is found to capture relatively accurately the flow phenomena and provide additional information regarding the produced flow structure dimensions, which is not available from the experiment.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Longfei Li〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉An efficient and accurate finite-element algorithm is described for the numerical solution of the incompressible Navier-Stokes (INS) equations. The new algorithm that solves the INS equations in a velocity-pressure reformulation is based on a split-step scheme in conjunction with the standard finite-element method. The split-step scheme employed for the temporal discretization of our algorithm completely separates the pressure updates from the solution of velocity variables. When the pressure equation is formed explicitly, the algorithm avoids solving a saddle-point problem; therefore, our algorithm has more flexibility in choosing finite-element spaces. In contrast, popular mixed finite-element methods that solve the INS equations in the primitive variables (or velocity-divergence formulation) lead to discrete saddle-point problems whose solution depends on the choice of finite-element spaces for velocity and pressure that is subject to the well-known Ladyzenskaja-Babuška-Brezzi (LBB) (or inf-sup) condition. For efficiency and robustness, Lagrange (piecewise-polynomial) finite elements of equal order for both velocity and pressure are used. Accurate numerical boundary condition for the pressure equation is also investigated. Motivated by a post-processing technique that calculates derivatives of a finite element solution with super-convergent error estimates, an alternative numerical boundary condition is proposed for the pressure equation at the discrete level. The new numerical pressure boundary condition that can be regarded as a better implementation of the compatibility boundary condition improves the boundary-layer errors of the pressure solution. A normal-mode analysis is performed using a simplified model problem on a uniform mesh to demonstrate the numerical properties of our methods. Convergence studies using 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉P〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msub〉〈/math〉 elements support the analytical results and demonstrates that our algorithm with the new numerical boundary condition achieves the optimal second-order accuracy for both velocity and pressure up-to the boundary. Benchmark problems are also computed and carefully compared with existing studies. Finally, as an example to illustrate that our approach can be easily adapted for higher-order finite elements, we solve the classical flow-past-a-cylinder problem using 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msub〉〈mrow〉〈mi mathvariant="double-struck"〉P〈/mi〉〈/mrow〉〈mrow〉〈mi〉n〈/mi〉〈/mrow〉〈/msub〉〈/math〉 finite elements with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈mi〉n〈/mi〉〈mo〉≥〈/mo〉〈mn〉1〈/mn〉〈/math〉.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Philipp Hähnel, Jakub Mareček, Julien Monteil, Fearghal O'Donncha〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Across numerous applications, forecasting relies on numerical solvers for partial differential equations (PDEs). Although the use of deep-learning techniques has been proposed, actual applications have been restricted by the fact the training data are obtained using traditional PDE solvers. Thereby, the uses of deep-learning techniques were limited to domains, where the PDE solver was applicable.〈/p〉 〈p〉We demonstrate a deep-learning framework for air-pollution monitoring and forecasting that provides the ability to train across different model domains, as well as a reduction in the run-time by two orders of magnitude. It presents a first-of-a-kind implementation that combines deep-learning and domain-decomposition techniques to allow model deployments extend beyond the domain(s) on which it has been trained.〈/p〉 〈/div〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): David Batista〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A Mimetic Finite Difference (MFD) implementation is proposed to solve the pressure-velocity system iteratively. The recursive procedure only solves for the velocity variable, while the pressure is calculated by direct substitution, simplifying the problem. The symmetric, positive definite system obtained is solved with an innovative implementation of a pre-conditioned, multilevel, conjugate gradient algorithm, where coarse algebraic objects are easily computed from the corresponding fine level ones, based on fundamental geometric properties of the MFD technology. A succinct eigen-analysis shows second order convergence on the velocity variable for Cartesian grids, which is corroborated numerically. Also, results consider different topologies and different medium properties, showing the applicability of the method and optimal convergence rates for the velocity on Cartesian and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉h〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉-uniform curved grids.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Kian Chuan Ong, Ming-Chih Lai〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We develop an immersed boundary projection method (IBPM) based on an unconditionally energy stable scheme to simulate the vesicle dynamics in a viscous fluid. Utilizing the block LU decomposition of the algebraic system, a novel fractional step algorithm is introduced by decoupling all solution variables, including the fluid velocity, fluid pressure, and the elastic tension. In contrast to previous works, the present method preserves both the fluid incompressibility and the interface inextensibility at a discrete level simultaneously. In conjunction with an implicit discretization of the bending force, the present method alleviates the time-step restriction, so the numerical stability is assured by non-increasing total discrete energy during the simulation. The numerical algorithm takes a linearithmic complexity by using preconditioned GMRES and FFT-based solvers. The grid convergence studies confirm the solution variables exhibit first-order convergence rate in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉-norm. We demonstrate the numerical results of the vesicle dynamics in a quiescent fluid, Poiseuille flow, and shear flow, which are congruent with the results in the literature.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Zekang Cheng, Jie Li, Ching Y. Loh, Li-Shi Luo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We present an interface conforming method for simulating two-dimensional and axisymmetric multiphase flows. In the proposed method, the interface is composed of straight segments which are part of mesh and move with the flow. This interface representation is an integral part of an Arbitrary Lagrangian-Eulerian (ALE) method on an moving adaptive unstructured mesh. Our principal aim is to develop an accurate and robust computational method for interfacial flows driven by strong surface tension and with weak viscous dissipation. We first construct discrete solutions satisfying the Laplace law on a circular/spherical interfaces exactly, 〈em〉i.e.〈/em〉, the balance between the surface tension and the pressure jump across an interface is achieved exactly in a discrete form. The accuracy and stability of these solutions are then investigated for a wide range of 〈em〉Ohnesorge〈/em〉 numbers, Oh. The dimensionless amplitude of the 〈em〉spurious current〈/em〉 is reduced to machine zero, 〈em〉i.e.〈/em〉, on the order of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mn〉10〈/mn〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉15〈/mn〉〈/mrow〉〈/msup〉〈/math〉 for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mtext〉Oh〈/mtext〉〈mo〉≥〈/mo〉〈msup〉〈mrow〉〈mn〉10〈/mn〉〈/mrow〉〈mrow〉〈mo linebreak="badbreak" linebreakstyle="after"〉−〈/mo〉〈mn〉3〈/mn〉〈/mrow〉〈/msup〉〈/math〉. Finally, the accuracy and capability of the proposed method are demonstrated through a series of benchmark tests with larger interface deformations. In particular, the method is validated with Prosperitti's analytic results of the bubble/drop oscillations and Peregrine's dripping faucet experiment, in which the values of Oh are small.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Hiroaki Nishikawa〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Simple modification techniques are proposed for making numerical fluxes amenable to unrealizable states (e.g., negative density) without degrading the design order of accuracy, so that a finite-volume solver never fails with unrealizable states arising in the solution reconstruction step and continues to run. The main idea is to evaluate quantities not affecting the order of accuracy but important for stabilization, e.g., a dissipation matrix, with low-order unreconstructed solutions. For the viscous flux, the viscosity is linearly extrapolated instead of being evaluated with linearly reconstructed temperatures to avoid a failure with a negative temperature. These ideas are quite general and may be applied to a wide range of numerical fluxes. In this paper, we illustrate them with the Roe flux and the alpha-damping viscous flux and demonstrate their effectiveness for cases, where a conventional technique encounters difficulties.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Tao Jiang, Jinlian Ren, Jinyun Yuan, Wen Zhou, Deng-Shan Wang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this work, a Lagrangian finite pointset model (FPM) is first developed to solve the 2D viscoelastic fluid governing equations, and then a corrected particle shifting technique (CPST) is introduced to eliminate the tensile instability in the long time simulations and added in the above FPM scheme (named as the FPMT). Subsequently, a coupled particle method (FPMT-SPH) is tentatively proposed to simulate the viscoelastic free surface flow, in which the FPMT method is used in the interior of fluid domain and the SPH method is adopted to treat the free surface near the boundary. The proposed FPMT method and FPMT-SPH method are motivated by: a) the spatial derivatives of the velocity and the viscoelastic stress are approximated and obtained by the weighted least squares method; b) the pressure is accurately solved by the application of projection method with a local iterative procedure; c) a corrected particle shifting technique is added to remedy the tensile instability (FPMT); d) an identification technique of free-surface particles is given in the FPMT-SPH method. The accuracy and the convergence of the proposed FPMT scheme for viscoelastic flow are first discussed by solving the planar flow based on the Oldroyd-B model, and compared with the analytical solutions. Secondly, the validity of the CPST is tested by several benchmarks, and compared with other numerical results. Thirdly, the viscoelastic lid-driven cavity flow at high Weissenberg number is simulated and compared with grid-based results, for further illustrating the robustness and the ability of the proposed FPMT method. Finally, the proposed coupled FPMT-SPH method is used to simulate the challenging free surface flow problem of a viscoelastic droplet impacting and spreading on rigid. All the numerical results show that the proposed particle method for the viscoelastic fluid or free surface flow is robust and reliable.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: Available online 21 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): R. Yamashita, L. Wutschitz, N. Nikiforakis〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper develops a full-field direct numerical simulation methodology of sonic boom in a stratified atmosphere. The entire flow field, ranging from the near field around a supersonic body to the far field extending to the ground, is modeled by the three-dimensional Euler equations with a gravitational source term. Thus far, it has been solved using a structured grid, and an application of previous simulation to complex geometries has been limited. In this study, we realize a full-field simulation by employing the following four numerical approaches: (i) a hierarchical structured adaptive mesh refinement (AMR) method, (ii) a Cartesian cut cell method, (iii) a well-balanced finite volume method, and (iv) a segmentation method of the computational domain. A new well-balanced, MUSCL-Hancock scheme applied over Cartesian AMR and cut cell grids for a stratified atmosphere is formulated. The computational results of an oblique shock wave in a stratified atmosphere agree well with the exact solution. A full-field simulation successfully reproduces the Drop test for Simplified Evaluation of Non-symmetrically Distributed sonic boom (D-SEND) #1, conducted by the Japan Aerospace Exploration Agency (JAXA). The results of this simulation are in good agreement with those of the previous computational study, the waveform parameter method, and flight test measurements. The grid convergence study shows that the mesh size is fine enough to assess pressure signatures over the entire flow field. These results demonstrate that a full-field simulation with AMR and cut cell grids is a powerful tool for extensively analyzing three-dimensional shock wave propagation in a stratified atmosphere.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Juan Manzanero, Gonzalo Rubio, David A. Kopriva, Esteban Ferrer, Eusebio Valero〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We present a provably stable discontinuous Galerkin spectral element method for the incompressible Navier–Stokes equations with artificial compressibility and variable density. Stability proofs, which include boundary conditions, that follow a continuous entropy analysis are provided. We define a mathematical entropy function that combines the traditional kinetic energy and an additional energy term for the artificial compressibility, and derive its associated entropy conservation law. The latter allows us to construct a provably stable split–form nodal Discontinuous Galerkin (DG) approximation that satisfies the summation–by–parts simultaneous–approximation–term (SBP–SAT) property. The scheme and the stability proof are presented for general curvilinear three–dimensional hexahedral meshes. We use the exact Riemann solver and the Bassi–Rebay 1 (BR1) scheme at the inter–element boundaries for inviscid and viscous fluxes respectively, and an explicit low storage Runge–Kutta RK3 scheme to integrate in time. We assess the accuracy and robustness of the method by solving a manufactured solution, the Kovasznay flow, a lid driven cavity, the inviscid Taylor–Green vortex, and the Rayleigh–Taylor instability.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): G. Chen, L. Chacón, L. Yin, B.J. Albright, D.J. Stark, R.F. Bird〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Conventional explicit electromagnetic particle-in-cell (PIC) algorithms do not conserve discrete energy exactly. Time-centered fully implicit PIC algorithms can conserve discrete energy exactly, but may introduce large dispersion errors in the light-wave modes. This can lead to intolerable simulation errors where accurate light propagation is needed (e.g. in laser-plasma interactions). In this study, we selectively combine the leap-frog and Crank-Nicolson methods to produce an exactly energy- and charge-conserving relativistic electromagnetic PIC algorithm. Specifically, we employ the leap-frog method for Maxwell's equations, and the Crank-Nicolson method for the particle equations. The semi-implicit formulation still features a timestep CFL, but facilitates exact global energy conservation, exact local charge conservation, and preserves the dispersion properties of the leap-frog method for the light wave. The algorithm employs a new particle pusher designed to maximize efficiency and minimize wall-clock-time impact vs. the explicit alternative. It has been implemented in a code named iVPIC, based on the Los Alamos National Laboratory VPIC code (〈a href="https://github.com/losalamos/vpic" target="_blank"〉https://github.com/losalamos/vpic〈/a〉). We present numerical results that demonstrate the properties of the scheme with sample test problems: relativistic two-stream instability, Weibel instability, and laser-plasma instabilities.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Hui Zheng, Chuanbing Zhou, Dong-Jia Yan, Yue-Sheng Wang, Chuanzeng Zhang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, the band structures of nanoscale phononic crystals based on the nonlocal elasticity theory are calculated by using a meshfree local radial basis function collocation method (LRBFCM). The direct method is applied to enhance the stability of the derivative calculations in the LRBFCM. A simple summation in the LRBFCM is proposed to deal with the integration related to the nonlocal stresses or tractions. The LRBFCM for the band structure calculations is validated by the results obtained with the first-principle and the transfer matrix (TM) method for one-dimensional (1D) phononic crystals, as well as the comparison of the frequency responses of the two-dimensional (2D) periodic structures.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Roger Käppeli, Dinshaw S. Balsara, Praveen Chandrashekar, Arijit Hazra〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉Discontinuous Galerkin (DG) methods have become mainstays in the accurate solution of hyperbolic systems, which suggests that they should also be important for computational electrodynamics (CED). Typically DG schemes are coupled with Runge-Kutta timestepping, resulting in RKDG schemes, which are also sometimes called DGTD schemes in the CED community. However, Maxwell's equations, which are solved in CED codes, have global mimetic constraints. In Balsara and Käppeli [〈em〉von Neumann Stability Analysis of Globally Constraint-Preserving DGTD and PNPM Schemes for the Maxwell Equations using Multidimensional Riemann Solvers〈/em〉, Journal of Computational Physics, 376 (2019) 1108-1137] the authors presented globally constraint-preserving DGTD schemes for CED. The resulting schemes had excellent low dissipation and low dispersion properties. Their one deficiency was that the maximal permissible CFL of DGTD schemes decreased with increasing order of accuracy. The goal of this paper is to show how this deficiency is overcome. Because CED entails the propagation of electromagnetic waves, we would also like to obtain DG schemes for CED that minimize dissipation and dispersion errors even more than the prior generation of DGTD schemes.〈/p〉 〈p〉Two recent advances make this possible. The first advance, which has been reported elsewhere, is the development of a multidimensional Generalized Riemann Problem (GRP) solver. The second advance relates to the use of Two Derivative Runge Kutta (TDRK) timestepping. This timestepping uses not just the solution of the multidimensional Riemann problem, it also uses the solution of the multidimensional GRP. When these two advances are melded together, we arrive at DG(TD)〈sup〉2〈/sup〉 schemes for CED, where the first “TD” stands for time-derivative and the second “TD” stands for the TDRK timestepping. The 〈em〉first goal〈/em〉 of this paper is to show how DG(TD)〈sup〉2〈/sup〉 schemes for CED can be formulated with the help of the multidimensional GRP and TDRK timestepping. The 〈em〉second goal〈/em〉 of this paper is to utilize the free parameters in TDRK timestepping to arrive at DG(TD)〈sup〉2〈/sup〉 schemes for CED that offer a uniformly large CFL with increasing order of accuracy while minimizing the dissipation and dispersion errors to exceptionally low values. The 〈em〉third goal〈/em〉 of this paper is to document a von Neumann stability analysis of DG(TD)〈sup〉2〈/sup〉 schemes so that their dissipation and dispersion properties can be quantified and studied in detail.〈/p〉 〈p〉At second order we find a DG(TD)〈sup〉2〈/sup〉 scheme with CFL of 0.25 and improved dissipation and dispersion properties; for a second order scheme. At third order we present a novel DG(TD)〈sup〉2〈/sup〉 scheme with CFL of 0.2571 and improved dissipation and dispersion properties; for a third order scheme. At fourth order we find a DG(TD)〈sup〉2〈/sup〉 scheme with CFL of 0.2322 and improved dissipation and dispersion properties. As an extra benefit, the resulting DG(TD)〈sup〉2〈/sup〉 schemes for CED require fewer synchronization steps on parallel supercomputers than comparable DGTD schemes for CED. We also document some test problems to show that the methods achieve their design accuracy.〈/p〉 〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Benedikt Dorschner, Ke Yu, Gianmarco Mengaldo, Tim Colonius〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We propose a mesh refinement technique for solving elliptic difference equations on unbounded domains based on the fast lattice Green's function (FLGF) method. The FLGF method exploits the regularity of the Cartesian mesh and uses the fast multipole method in conjunction with fast Fourier transforms to yield linear complexity and decrease time-to-solution. We extend this method to a multi-resolution scheme and allow for locally refined Cartesian blocks embedded in the computational domain. Appropriately chosen interpolation and regularization operators retain consistency between the discrete Laplace operator and its inverse on the unbounded domain. Second-order accuracy and linear complexity are maintained, while significantly reducing the number of degrees of freedom and hence the computational cost.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): David Gunderman, Natasha Flyer, Bengt Fornberg〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This work presents a numerical algorithm for using radial basis function-generated finite differences (RBF-FD) to solve partial differential equations (PDEs) on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉S〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 using polyharmonic splines with added polynomials defined in a 2D plane (PHS+Poly). We introduce a novel method for calculating RBF-FD PHS+Poly differentiation weights on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉S〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 using first a Householder reflection and then a projection onto the tangent plane. The new PHS+Poly RBF-FD method is implemented on two standard test cases: 1) solid body rotation on 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉S〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉 and 2) 3D tracer transport within Earth's atmosphere. Compared to existing methods (including those in the RBF literature) at similar resolutions, this approach requires fewer degrees of freedom and is algorithmically much simpler. A MATLAB code to implement the method is included in the Appendix.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Daniil Bochkov, Frederic Gibou〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We present a simple numerical algorithm for solving elliptic equations where the diffusion coefficient, the source term, the solution and its flux are discontinuous across an irregular interface. The algorithm produces second-order accurate solutions and first-order accurate gradients in the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mo〉∞〈/mo〉〈/mrow〉〈/msup〉〈/math〉-norm on Cartesian grids. The condition number is bounded, regardless of the ratio of the diffusion constant and scales like that of the standard 5-point stencil approximation on a rectangular grid with no interface. Numerical examples are given in two and three spatial dimensions.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Jiuhua Hu, Guanglian Li〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We consider the initial boundary value problem for the time-fractional diffusion equation with a homogeneous Dirichlet boundary condition and an inhomogeneous initial data 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈mi〉a〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉∈〈/mo〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉D〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 in a bounded domain 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mi〉D〈/mi〉〈mo〉⊂〈/mo〉〈msup〉〈mrow〉〈mi mathvariant="double-struck"〉R〈/mi〉〈/mrow〉〈mrow〉〈mi〉d〈/mi〉〈/mrow〉〈/msup〉〈/math〉 with a sufficiently smooth boundary. We analyze the homogenized solution under the assumption that the diffusion coefficient 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si26.svg"〉〈msup〉〈mrow〉〈mi〉κ〈/mi〉〈/mrow〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉x〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 is smooth and periodic with the period 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi〉ϵ〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉0〈/mn〉〈/math〉 being sufficiently small. We derive that its first order approximation measured by both pointwise-in-time in 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si5.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉D〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si188.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mi〉p〈/mi〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mi〉θ〈/mi〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo〉;〈/mo〉〈msup〉〈mrow〉〈mi〉H〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉(〈/mo〉〈mi〉D〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 for 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si170.svg"〉〈mi〉p〈/mi〉〈mo〉∈〈/mo〉〈mo stretchy="false"〉[〈/mo〉〈mn〉1〈/mn〉〈mo〉,〈/mo〉〈mo〉∞〈/mo〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si126.svg"〉〈mi〉θ〈/mi〉〈mo〉∈〈/mo〉〈mo stretchy="false"〉(〈/mo〉〈mn〉0〈/mn〉〈mo〉,〈/mo〉〈mi〉T〈/mi〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 has a convergence rate of 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si281.svg"〉〈mi mathvariant="script"〉O〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo stretchy="false"〉/〈/mo〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 when the dimension 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si10.svg"〉〈mi〉d〈/mi〉〈mo〉≤〈/mo〉〈mn〉2〈/mn〉〈/math〉 and 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si11.svg"〉〈mi mathvariant="script"〉O〈/mi〉〈mo stretchy="false"〉(〈/mo〉〈msup〉〈mrow〉〈mi〉ϵ〈/mi〉〈/mrow〉〈mrow〉〈mn〉1〈/mn〉〈mo stretchy="false"〉/〈/mo〉〈mn〉6〈/mn〉〈/mrow〉〈/msup〉〈mo stretchy="false"〉)〈/mo〉〈/math〉 when 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si12.svg"〉〈mi〉d〈/mi〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉3〈/mn〉〈/math〉. Several numerical tests are presented to demonstrate the performance of the first order approximation.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Naveen Kumar Garg, N.H. Maruthi, S.V. Raghurama Rao, M. Sekhar〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉In this study, convection-pressure split Euler flux functions which contain weakly hyperbolic convective subsystems are analyzed. A system of first-order partial differential equations (PDEs) is said to be weakly hyperbolic if the corresponding flux Jacobian does not contain a complete set of linearly independent (LI) eigenvectors. Thus, the application of existing flux difference splitting (FDS) based schemes, which depend heavily on both eigenvalues and eigenvectors, are non-trivial to such systems. In the case of weakly hyperbolic systems, a required set of LI eigenvectors can be constructed through the addition of generalized eigenvectors by utilizing the theory of Jordan canonical forms. Once this is achieved for a weakly hyperbolic convective subsystem, an upwind solver can be constructed in the splitting framework.〈/p〉 〈p〉In the present work, the above approach is used for developing two new schemes. The first scheme is based on the Zha–Bilgen type splitting while the second is based on the Toro–Vázquez splitting. Both the schemes are tested on various benchmark problems in one-dimension (1-D) and two-dimensions (2-D). The concept of generalized eigenvectors based on Jordan forms is found to be useful in dealing with the weakly hyperbolic parts of the considered Euler systems.〈/p〉 〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Sriramkrishnan Muralikrishnan, Tan Bui-Thanh, John N. Shadid〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We propose a multilevel approach for trace systems resulting from hybridized discontinuous Galerkin (HDG) methods. The key is to blend ideas from nested dissection, domain decomposition, and high-order characteristic of HDG discretizations. Specifically, we first create a coarse solver by eliminating and/or limiting the front growth in nested dissection. This is accomplished by projecting the trace data into a sequence of same or high-order polynomials on a set of increasingly 〈em〉h〈/em〉-coarser edges/faces. We then combine the coarse solver with a block-Jacobi fine scale solver to form a two-level solver/preconditioner. Numerical experiments indicate that the performance of the resulting two-level solver/preconditioner depends on the smoothness of the solution and can offer significant speedups and memory savings compared to the nested dissection direct solver. While the proposed algorithms are developed within the HDG framework, they are applicable to other hybrid(ized) high-order finite element methods. Moreover, we show that our multilevel algorithms can be interpreted as a multigrid method with specific intergrid transfer and smoothing operators. With several numerical examples from Poisson, pure transport, and convection-diffusion equations we demonstrate the robustness and scalability of the algorithms with respect to solution order. While scalability with mesh size in general is not guaranteed and depends on the smoothness of the solution and the type of equation, improving it is a part of future work.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Wanai Li, Qian Wang, Yu-Xin Ren〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper presents an accuracy-preserving 〈em〉p〈/em〉-weighted limiter for discontinuous Galerkin methods on one-dimensional and two-dimensional triangular grids. The 〈em〉p〈/em〉-weighted limiter is the extension of the second-order WENO limiter by Li et al. (2018) [22] to high-order accuracy, with the following important improvements of the limiting procedure. First, the candidate polynomials of the 〈em〉p〈/em〉-weighted limiter are the 〈em〉p〈/em〉-hierarchical orthogonal polynomials of the current cell, and the linear polynomials constructed by minimizing the projection error on the face-neighboring cells. Second, the 〈em〉p〈/em〉-weighted procedure introduces a new smoothness indicator which has less numerical dissipation comparing with the classical WENO one. The smoothness indicator is efficiently computed through a quadrature-free approach that takes advantage of the orthogonal property of the basis functions. Third, the small positive number 〈em〉ϵ〈/em〉, which is introduced in the weights to avoid dividing by zero, is set as a function of the smoothness indicator to preserve accuracy near smooth extremas. Numerous benchmark problems are solved to test the 〈em〉p〈/em〉1, 〈em〉p〈/em〉3 and 〈em〉p〈/em〉5 discontinuous Galerkin schemes using the 〈em〉p〈/em〉-weighted limiter. Numerical results demonstrate that the 〈em〉p〈/em〉-weighted limiter is capable of capturing strong shocks while preserving accuracy in smooth regions.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Emil Klahn, Holger Grosshans〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The agglomeration of particles during the handling of powders results in caking, lumping or the local accumulation of electrostatic energy which represents a serious hazard to the operational safety of industrial facilities. In the case of dry powders the attraction in-between particles can be mainly attributed to van der Waals and electrostatic forces. Nonetheless, due to the challenges related to the small size and distance of relevant particles and the optical density of powder flows the detailed physical mechanisms of their interaction are so far little investigated. In this paper we present a novel numerical approach which is based on an algorithm developed by Erleben [1] in the field of computer graphics. This algorithm is extended to compute binary and multiple particle interaction with each other and solid surfaces. Therein, besides van der Waals and electrostatic forces also collisional forces and plastic particle deformation is accounted for. The herein presented results demonstrate that this algorithm allows to predict accurately and efficiently whether particles agglomerate or separate depending on their kinetic parameters. In particular, the imposed constrain forces prevent spurious velocity fluctuations and potential particle overlapping in statically overdetermined systems. Simulated test cases reveal how electrostatic and van der Waals forces lead to the growth of structures in case the particle restitution coefficient is sufficiently low.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Mathis Fricke, Tomislav Marić, Dieter Bothe〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉We consider the interface advection problem by a prescribed velocity field in the special case when the interface intersects the domain boundary, i.e. in the presence of a contact line. This problem emerges from the discretization of continuum models for dynamic wetting. The 〈em〉kinematic evolution equation〈/em〉 for the dynamic contact angle (Fricke et al., 2019) expresses the fundamental relationship between the rate of change of the contact angle and the structure of the transporting velocity field. The goal of the present work is to develop an interface advection method that is consistent with the fundamental kinematics and transports the contact angle correctly with respect to a prescribed velocity field. In order to verify the advection method, the kinematic evolution equation is solved numerically and analytically (for special cases).〈/p〉 〈p〉We employ the geometrical Volume-of-Fluid (VOF) method on a structured Cartesian grid to solve the hyperbolic transport equation for the interface in two spatial dimensions. We introduce generalizations of the Youngs and ELVIRA methods to reconstruct the interface close to the domain boundary. Both methods deliver first-order convergent results for the motion of the contact line. However, the Boundary Youngs method shows strong oscillations in the numerical contact angle that do not converge with mesh refinement. In contrast to that, the Boundary ELVIRA method provides linear convergence of the numerical contact angle transport.〈/p〉 〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): François P. Hamon, Martin Schreiber, Michael L. Minion〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉The modeling of atmospheric processes in the context of weather and climate simulations is an important and computationally expensive challenge. The temporal integration of the underlying PDEs requires a very large number of time steps, even when the terms accounting for the propagation of fast atmospheric waves are treated implicitly. Therefore, the use of parallel-in-time integration schemes to reduce the time-to-solution is of increasing interest, particularly in the numerical weather forecasting field.〈/p〉 〈p〉We present a multi-level parallel-in-time integration method combining the Parallel Full Approximation Scheme in Space and Time (PFASST) with a spatial discretization based on Spherical Harmonics (SH). The iterative algorithm computes multiple time steps concurrently by interweaving parallel high-order fine corrections and serial corrections performed on a coarsened problem. To do that, we design a methodology relying on the spectral basis of the SH to coarsen and interpolate the problem in space.〈/p〉 〈p〉The methods are evaluated on the shallow-water equations on the sphere using a set of tests commonly used in the atmospheric flow community. We assess the convergence of PFASST-SH upon refinement in time. We also investigate the impact of the coarsening strategy on the accuracy of the scheme, and specifically on its ability to capture the high-frequency modes accumulating in the solution. Finally, we study the computational cost of PFASST-SH to demonstrate that our scheme resolves the main features of the solution multiple times faster than the serial schemes.〈/p〉 〈/div〉

Beschreibung: 〈p〉Publication date: 1 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 406〈/p〉 〈p〉Author(s): Hui Guo, Xinyuan Liu, Yang Yang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉In this paper, we develop high-order bound-preserving (BP) finite difference (FD) methods for the coupled system of compressible miscible displacements. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the 〈em〉j〈/em〉th component, 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈/math〉, should be between 0 and 1. It is well known that 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈/math〉 does not satisfy a maximum-principle. Hence most of the existing BP techniques cannot be applied directly. The main idea in this paper is to construct the positivity-preserving techniques to all 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈msubsup〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈/mrow〉〈mrow〉〈mo〉′〈/mo〉〈/mrow〉〈/msubsup〉〈mi〉s〈/mi〉〈/math〉 and enforce 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si3.svg"〉〈msub〉〈mrow〉〈mo〉∑〈/mo〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈msub〉〈mrow〉〈mi〉c〈/mi〉〈/mrow〉〈mrow〉〈mi〉j〈/mi〉〈/mrow〉〈/msub〉〈mo linebreak="goodbreak" linebreakstyle="after"〉=〈/mo〉〈mn〉1〈/mn〉〈/math〉 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si4.svg"〉〈mi〉d〈/mi〉〈mi〉p〈/mi〉〈mo stretchy="false"〉/〈/mo〉〈mi〉d〈/mi〉〈mi〉t〈/mi〉〈/math〉 as a source in the concentration equation and choose suitable “consistent” numerical fluxes in the pressure and concentration equations. Recently, the high-order BP discontinuous Galerkin (DG) methods for miscible displacements were introduced in [4]. However, the BP technique for DG methods is not straightforward extendable to high-order FD schemes. There are two main difficulties. Firstly, it is not easy to determine the time step size in the BP technique. In finite difference schemes, we need to choose suitable time step size first and then apply the flux limiter to the numerical fluxes. Subsequently, we can compute the source term in the concentration equation, leading to a new time step constraint that may not be satisfied by the time step size applied in the flux limiter. Therefore, it would be very difficult to determine how large the time step is. Secondly, the general treatment for the diffusion term, e.g. centered difference, in miscible displacements may require a stencil whose size is larger than that for the convection term. It would be better to construct a new spatial discretization for the diffusion term such that a smaller stencil can be used. In this paper, we will solve both problems. We first construct a special discretization of the convection term, which yields the desired approximations of the source. Then we can find out the time step size that suitable for the BP technique and apply the flux limiters. Moreover, we will also construct a special algorithm for the diffusion term whose stencil is the same as that used for the convection term. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: Available online 21 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Morten Jakobsen, Ru-Shan Wu, Xingguo Huang〈/p〉

Beschreibung: 〈p〉Publication date: Available online 21 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): K. Brenner, R. Masson, E.H. Quenjel〈/p〉

Beschreibung: 〈p〉Publication date: Available online 21 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Thomas Milcent, Antoine Lemoine〈/p〉

Beschreibung: 〈p〉Publication date: Available online 19 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Yingzhou Li, Jianfeng Lu, Anqi Mao〈/p〉

Beschreibung: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 409〈/p〉 〈p〉Author(s): Kejun Tang, Qifeng Liao〈/p〉

Beschreibung: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 409〈/p〉 〈p〉Author(s): Gregory Lecrivain, Taisa Beatriz Pacheco Grein, Ryoichi Yamamoto, Uwe Hampel, Takashi Taniguchi〈/p〉

Beschreibung: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 409〈/p〉 〈p〉Author(s): D. Faghihi, V. Carey, C. Michoski, R. Hager, S. Janhunen, C.S. Chang, R.D. Moser〈/p〉

Beschreibung: 〈p〉Publication date: Available online 10 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Fabien Lespagnol, Gautier Dakin〈/p〉

Beschreibung: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 409〈/p〉 〈p〉Author(s): Li Luo, Lulu Liu, Xiao-Chuan Cai, David E. Keyes〈/p〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Bruno Després, Hervé Jourdren〈/p〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Jordi Feliu-Fabà, Yuwei Fan, Lexing Ying〈/p〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Ounan Ding, Tamar Shinar, Craig Schroeder〈/p〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Stanislav Harizanov, Raytcho Lazarov, Svetozar Margenov, Pencho Marinov, Joseph Pasciak〈/p〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Matthew R. New-Tolley, Mikhail N. Shneider, Richard B. Miles〈/p〉

Beschreibung: 〈p〉Publication date: Available online 5 February 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Niccolò Discacciati, Jan S. Hesthaven, Deep Ray〈/p〉

Beschreibung: 〈p〉Publication date: 15 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 409〈/p〉 〈p〉Author(s): Nathaniel Trask, Pavel Bochev, Mauro Perego〈/p〉

Beschreibung: 〈p〉Publication date: 1 May 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 408〈/p〉 〈p〉Author(s): Yulin Pan〈/p〉

Beschreibung: 〈p〉Publication date: Available online 21 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Benedict Dingfelder, Florian J. Hindenlang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉In magnetized plasmas of fusion devices the strong magnetic field leads to highly anisotropic physics where solution scales along field lines are much larger than perpendicular to it. Hence, regarding both accuracy and efficiency, a numerical method should allow to address parallel and perpendicular resolutions independently. In this work, we consider the eigenvalue problem of a two-dimensional anisotropic wave equation with variable coefficients which is a simplified model of linearized ideal magnetohydrodynamics.〈/p〉 〈p〉For this, we propose to use a mesh that is aligned with the magnetic field and choose to discretize the problem with a discontinuous Galerkin method which naturally allows for non-conforming interfaces.〈/p〉 〈p〉First, we analyze the eigenvalue spectrum of a constant coefficient anisotropic wave equation, and demonstrate that this approach improves the accuracy by up to seven orders of magnitude, if compared to a non-aligned method with the same number of degrees of freedom. In particular, the results improve for eigenfunctions with high mode numbers.〈/p〉 〈p〉We also apply the method to compute the eigenvalue spectrum of the associated anisotropic wave equation with variable coefficients of flux surfaces of a Stellarator configuration. We benchmark the results against a spectral code.〈/p〉 〈/div〉

Beschreibung: 〈p〉Publication date: Available online 21 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Alex Gorodetsky, Gianluca Geraci, Michael Eldred, John D. Jakeman〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We describe and analyze a variance reduction approach for Monte Carlo (MC) sampling that accelerates the estimation of statistics of computationally expensive simulation models using an ensemble of models with lower cost. These lower cost models — which are typically lower fidelity with unknown statistics — are used to reduce the variance in statistical estimators relative to a MC estimator with equivalent cost. We derive the conditions under which our proposed approximate control variate framework recovers existing multifidelity variance reduction schemes as special cases. We demonstrate that existing recursive/nested strategies are suboptimal because they use the additional low-fidelity models only to efficiently estimate the unknown mean of the first low-fidelity model. As a result, they cannot achieve variance reduction beyond that of a control variate estimator that uses a single low-fidelity model with known mean. However, there often exists about an order-of-magnitude gap between the maximum achievable variance reduction using all low-fidelity models and that achieved by a single low-fidelity model with known mean. We show that our proposed approach can exploit this gap to achieve greater variance reduction by using non-recursive sampling schemes. The proposed strategy reduces the total cost of accurately estimating statistics, especially in cases where only low-fidelity simulation models are accessible for additional evaluations. Several analytic examples and an example with a hyperbolic PDE describing elastic wave propagation in heterogeneous media are used to illustrate the main features of the methodology.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: Available online 13 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Florian Dugast, Yann Favennec, Christophe Josset〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This paper presents a topology optimization algorithm based on the lattice Boltzmann method coupled with a level-set method for increasing the efficiency of reactive fluid flows. The multi-relaxation time model is considered for the lattice Boltzmann collision operator, allowing higher Reynolds numbers flow simulations compared to the ordinary single-relaxation time model. The cost function gradient is obtained with the derivation of the adjoint-state formulation for the fully coupled problem. The proposed method is tested successfully on several numerical applications involving Reynolds numbers from 10 up to 1,000, as well as with different Damkohler and Peclet numbers. A limitation of the maximal pressure drop is also applied. The obtained results demonstrate that the proposed numerical method is robust and efficient for solving topology optimization problems of reactive fluid flows, in different operating conditions.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Amaresh Sahu, Yannick A.D. Omar, Roger A. Sauer, Kranthi K. Mandadapu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉An arbitrary Lagrangian–Eulerian (ALE) finite element method for arbitrarily curved and deforming two-dimensional materials and interfaces is presented here. An ALE theory is developed by endowing the surface with a mesh whose in-plane velocity need not depend on the in-plane material velocity, and can be specified arbitrarily. A finite element implementation of the theory is formulated and applied to curved and deforming surfaces with in-plane incompressible flows. Numerical inf–sup instabilities associated with in-plane incompressibility are removed by locally projecting the surface tension onto a discontinuous space of piecewise linear functions. The general isoparametric finite element method, based on an arbitrary surface parametrization with curvilinear coordinates, is tested and validated against several numerical benchmarks. A new physical insight is obtained by applying the ALE developments to cylindrical fluid films, which are computationally and analytically found to be stable to non-axisymmetric perturbations, and unstable with respect to long-wavelength axisymmetric perturbations when their length exceeds their circumference. A Lagrangian scheme is attained as a special case of the ALE formulation. Though unable to model fluid films with sustained shear flows, the Lagrangian scheme is validated by reproducing the cylindrical instability. However, relative to the ALE results, the Lagrangian simulations are found to have spatially unresolved regions with few nodes, and thus larger errors.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Eric Cancès, Virginie Ehrlacher, Frédéric Legoll, Benjamin Stamm, Shuyang Xiang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉This contribution is the numerically oriented companion article of the work [9]. We focus here on the numerical resolution of the embedded corrector problem introduced in [8], [9] in the context of homogenization of diffusion equations. Our approach consists in considering a corrector-type problem, posed on the whole space, but with a diffusion matrix which is constant outside some bounded domain. In [9], we have shown how to define three approximate homogenized diffusion coefficients on the basis of the embedded corrector problem. We have also proved that these approximations all converge to the exact homogenized coefficients when the size of the bounded domain increases.〈/p〉 〈p〉We show here that, under the assumption that the diffusion matrix is piecewise constant, the corrector problem to solve can be recast as an integral equation. In case of spherical inclusions with isotropic materials, we explain how to efficiently discretize this integral equation using spherical harmonics, and how to use the fast multipole method (FMM) to compute the resulting matrix-vector products at a cost which scales only linearly with respect to the number of inclusions. Numerical tests illustrate the performance of our approach in various settings.〈/p〉 〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Pierre Lallemand, Li-Shi Luo〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This work combines the lattice Boltzmann equation (LBE) and the overset method to simulate moving boundary problems in Navier-Stokes flows in two dimensions (2D). The transformation of the velocity moments of the distribution functions between a moving frame of reference and the one at rest is analyzed. The flow past a cylinder moving with a prescribed motion is used to validate the proposed LBE-overset method. We show that the proposed LBE-overset method does reduce the numerical noise generated by the relative motion between a moving object and the underlying Eulerian mesh for flow fields by several orders of magnitude.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Nicolò Scapin, Pedro Costa, Luca Brandt〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We present a numerical method for interface-resolved simulations of evaporating two-fluid flows based on the volume-of-fluid (VoF) method. The method has been implemented in an efficient FFT-based two-fluid Navier-Stokes solver, using an algebraic VoF method for the interface representation, and extended with the transport equations of thermal energy and vaporized liquid mass for the single-component evaporating liquid in an inert gas. The conservation of vaporizing liquid and computation of the interfacial mass flux are performed with the aid of a reconstructed signed-distance field, which enables the use of well-established methods for phase change solvers based on level-set methods. The interface velocity is computed with a novel approach that ensures accurate mass conservation, by constructing a divergence-free extension of the liquid velocity field onto the entire domain. The resulting approach does not depend on the type of interface reconstruction (i.e. can be employed in both algebraic and geometrical VoF methods). We extensively verified and validated the overall method against several benchmark cases, and demonstrated its excellent mass conservation and good overall performance for simulating evaporating two-fluid flows in two and three dimensions.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): André Garon, Michel C. Delfour〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉In this paper, the efficiency in mesh updating (r-adaptivity) of the 〈em〉Transfinite Mean value Interpolation〈/em〉 (TMI) and its generalization (〈em〉k〈/em〉-TMI) are compared on three standardized problems to the well-known 〈em〉Inverse Distance Weighted interpolation〈/em〉 (IDW) and 〈em〉Radial Basis Function interpolation〈/em〉 (RBF) for unstructured data points and the new 〈em〉k-Transfinite Barycentric Interpolation〈/em〉 (〈em〉k〈/em〉-TBI) for structured data points such as, for instance, curves or surfaces in 3D. This is achieved by introducing a dynamical version of these interpolations via an ordinary differential equation that can be solved by standard ODE methods that are more economical than, for instance, solving vector partial differential equations as in the 〈em〉pseudo-solid method〈/em〉.〈/p〉 〈p〉A review of the very recent mathematical foundations of the 〈em〉k〈/em〉-TMI and 〈em〉k〈/em〉-TBI constructed from the function alone (standard) or from the function and its derivatives (enhanced) is provided in the first part of the paper.〈/p〉 〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Mauricio Ponga, Kaushik Bhattacharya, Michael Ortiz〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We present a novel methodology to compute relaxed dislocations core configurations, and their energies in crystalline metallic materials using large-scale 〈em〉ab-intio〈/em〉 simulations. The approach is based on MacroDFT, a coarse-grained density functional theory method that accurately computes the electronic structure with sub-linear scaling resulting in a tremendous reduction in cost. Due to its implementation in 〈em〉real-space〈/em〉, MacroDFT has the ability to harness petascale resources to study materials and alloys through accurate 〈em〉ab-initio〈/em〉 calculations. Thus, the proposed methodology can be used to investigate dislocation cores and other defects where long range elastic effects play an important role, such as in dislocation cores, grain boundaries and near precipitates in crystalline materials. We demonstrate the method by computing the relaxed dislocation cores in prismatic dislocation loops and dislocation segments in magnesium (Mg). We also study the interaction energy with a line of Aluminum (Al) solutes. Our simulations elucidate the essential coupling between the quantum mechanical aspects of the dislocation core and the long range elastic fields that they generate. In particular, our quantum mechanical simulations are able to describe the logarithmic divergence of the energy in the far field as is known from classical elastic theory. In order to reach such scaling, the number of atoms in the simulation cell has to be exceedingly large, and cannot be achieved with the 〈em〉state-of-the-art〈/em〉 density functional theory implementations.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Eduard Ubeda, Ivan Sekulic, Juan M. Rius, Alex Heldring〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Recent implementations of the Electric-Field Integral Equation (EFIE) for the electromagnetic scattering analysis of perfectly conducting targets rely on the electric current expansion with the monopolar-RWG basis functions, discontinuous across mesh edges, and the field testing over volumetric subdomains attached to the surface boundary triangulation. As compared to the standard RWG-based EFIE-approaches, normally continuous across edges, these schemes exhibit enhanced versatility, allowing the analysis of geometrically non-conformal meshes, and improved accuracy, especially for subwavelength sharp-edged conductors. In this paper, we present a monopolar-RWG discretization by the Method of Moments (MoM) of the Combined-Field Integral Equation (CFIE) resulting from the addition of a volumetrically tested discretization of the EFIE and the Galerkin tested MFIE-implementation. We show for sharp-edged conductors the degree of improved accuracy in the computed RCS and the convergence properties in the iterative search of the solution. More importantly, as we show in the paper, these implementations become in practice advantageous because of their robustness to flaws in the grid generation or their agility in handling complex meshes arising from the interconnection of independently meshed domains. The hybrid RWG/monopolar-RWG discretization of the CFIE defines the RWG discretization over geometrically conformal and smoothly varying mesh regions and inserts the monopolar-RWG expansion strictly at sharp edges, for improved accuracy purposes, or over boundary lines between partitioning mesh domains, for the sake of enhanced versatility. These hybrid schemes offer similar accuracy as their fully monopolar-RWG counterparts but with fewer unknowns and allow naturally non-conformal mesh transitions without inserting additional inter-domain continuity conditions or new artificial currents.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): T. Corot, P. Hoch, E. Labourasse〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We describe an Arbitrary-Lagrangian-Eulerian (ALE) method for the compressible Euler system with capillary force. The algorithm is split in two steps. First, the Lagrangian step is based on cell-centred schemes [9], [20], [46]. The surface tension force is discretized in order to exactly verify the Laplace law at the discrete level. We also provide a second-order spatial extension and a low-Mach correction, which do not break the well-balanced property of the scheme. The Lagrangian scheme is assessed on several problems, particularly on a linear Richtmyer-Meshkov instability which is the targeted application. The second step is the rezoning and remapping done thanks to a swept-region method using exact intersections near the interface. We use a Volume Of Fluid (VOF) method to track the interface. We describe the treatment of mixed-cells, and in particular the thermodynamics closure and the curvature calculation. The new scheme is used to investigate the influence of surface tension on a non-linear Richtmyer-Meshkov instability.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Neeraj Sarna, Harshit Kapadia, Manuel Torrilhon〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Previous works have developed boundary conditions that lead to the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉-boundedness of solutions to the linearised moment equations. Here we present a spatial discretization that preserves the 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msup〉〈mrow〉〈mi〉L〈/mi〉〈/mrow〉〈mrow〉〈mn〉2〈/mn〉〈/mrow〉〈/msup〉〈/math〉-stability by recovering integration-by-parts over the discretized domain and by imposing boundary conditions using a simultaneous-approximation-term (SAT). We develop three different forms of the SAT using: (i) characteristic splitting of moment equation's boundary conditions; (ii) decoupling of moments in moment equations; and (iii) characteristic splitting of Boltzmann equation's boundary conditions. We discuss how the first two forms differ in terms of their usage and implementation. We show that the third form is equivalent to using an upwind kinetic numerical flux along the boundary, and we argue that even though it provides stability, it prescribes the incorrect number of boundary conditions. Using benchmark problems, we compare the accuracy of moment solutions computed using different SATs. Our numerical experiments also provide new insights into the convergence of moment approximations to the Boltzmann equation's solution.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Yanni Gao, Guangwei Yuan, Shuai Wang, Xudeng Hang〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We construct a finite volume element scheme with a monotonicity correction for anisotropic diffusion problems on general quadrilateral meshes, where the strict convexity restriction on the meshes is removed. The main contributions of this paper include three aspects. Firstly, the classical finite volume element (C-FVE) method is extended to severely distorted quadrilateral meshes even with concave cells by virtue of a new overlapping dual partition and a new gradient approximation. In fact, the choice of this dual partition and gradient approximation is also conducive to the construction of a monotone scheme. Secondly, a new monotonicity correction is suggested, based on which we obtain a monotone finite volume element (M-FVE) method. The resulting M-FVE method still keeps the local conservation and is easy for implementation. Finally, we analyze theoretically the truncation error and the monotonicity for this scheme. Besides, the existence of a solution to this nonlinear scheme is proved by applying the Brouwer's fixed point theorem. Numerical results demonstrate that the M-FVE method has the approximate second-order accuracy and preserves well the positivity of the solution for both isotropic and anisotropic diffusion problems on severely distorted quadrilateral meshes.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Tom H. Anderson, Benjamin J. Civiletti, Peter B. Monk, Akhlesh Lakhtakia〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉A design tool was formulated for optimizing the efficiency of inorganic, thin-film, photovoltaic solar cells. The solar cell can have multiple semiconductor layers in addition to antireflection coatings, passivation layers, and buffer layers. The solar cell is backed by a metallic grating which is periodic along a fixed direction. The rigorous coupled-wave approach is used to calculate the electron-hole-pair generation rate. The hybridizable discontinuous Galerkin method is used to solve the drift-diffusion equations that govern charge-carrier transport in the semiconductor layers. The chief output is the solar-cell efficiency which is maximized using the differential evolution algorithm to determine the optimal dimensions and bandgaps of the semiconductor layers.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Wei Su, Lianhua Zhu, Peng Wang, Yonghao Zhang, Lei Wu〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉One of the central problems in the study of rarefied gas dynamics is to find the steady-state solution of the Boltzmann equation quickly. When the Knudsen number is large, i.e. the system is highly rarefied, the conventional iterative scheme can lead to convergence within a few iterations. However, when the Knudsen number is small, i.e. the flow falls in the near-continuum regime, hundreds of thousands iterations are needed, and yet the “converged” solutions are prone to be contaminated by accumulated error and large numerical dissipation. Recently, based on the gas kinetic models, the implicit unified gas kinetic scheme (UGKS) and its variants have significantly reduced the number of iterations in the near-continuum flow regime, but still much higher than that of the highly rarefied gas flows. In this paper, we put forward a general synthetic iterative scheme (GSIS) to find the steady-state solutions of rarefied gas flows within dozens of iterations at any Knudsen number. The key ingredient of our scheme is that the macroscopic equations, which are solved together with the Boltzmann equation and help to adjust the velocity distribution function, not only asymptotically preserve the Navier-Stokes limit in the framework of Chapman-Enskog expansion, but also contain the Newton's law for stress and the Fourier's law for heat conduction explicitly. For this reason, like the implicit UGKS, the constraint that the spatial cell size should be smaller than the mean free path of gas molecules is removed, but we do not need the complex evaluation of numerical flux at cell interfaces. What's more, as the GSIS does not rely on the specific collision operator, it can be naturally extended to quickly find converged solutions for mixture flows and even flows involving chemical reactions. These two superior advantages are expected to accelerate the slow convergence in the simulation of near-continuum flows via the direct simulation Monte Carlo method and its low-variance version.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: Available online 8 January 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics〈/p〉 〈p〉Author(s): Michael P. Adams, Marvin L. Adams, W. Daryl Hawkins, Timmie Smith, Lawrence Rauchwerger, Nancy M. Amato, Teresa S. Bailey, Robert D. Falgout, Adam Kunen, Peter Brown〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We have found provably optimal algorithms for full-domain discrete-ordinate transport sweeps on a class of grids in 2D and 3D Cartesian geometry that are regular at a coarse level but arbitrary within the coarse blocks. We describe these algorithms and show that they always execute the full eight-octant (or four-quadrant if 2D) sweep in the minimum possible number of stages for a given 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mi〉x〈/mi〉〈/mrow〉〈/msub〉〈mo〉×〈/mo〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mi〉y〈/mi〉〈/mrow〉〈/msub〉〈mo〉×〈/mo〉〈msub〉〈mrow〉〈mi〉P〈/mi〉〈/mrow〉〈mrow〉〈mi〉z〈/mi〉〈/mrow〉〈/msub〉〈/math〉 partitioning. Computational results confirm that our optimal scheduling algorithms execute sweeps in the minimum possible stage count. Observed parallel efficiencies agree well with our performance model. Our PDT transport code has achieved approximately 68% parallel efficiency with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mo linebreak="badbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1.5〈/mn〉〈mi〉M〈/mi〉〈/math〉 parallel threads, relative to 8 threads, on a simple weak-scaling problem with only three energy groups, 10 direction per octant, and 4096 cells/thread. Our ARDRA code has achieved 71% efficiency with 〈math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.svg"〉〈mo linebreak="badbreak" linebreakstyle="after"〉〉〈/mo〉〈mn〉1.5〈/mn〉〈mi〉M〈/mi〉〈/math〉 cores, relative to 16 cores, with 36 directions per octant and 48 energy groups. We demonstrate similar efficiencies with PDT on a realistic set of nuclear-reactor test problems, with unstructured meshes that resolve fine geometric details. These results demonstrate that discrete-ordinates transport sweeps can be executed with high efficiency using more than 10〈sup〉6〈/sup〉 parallel processes.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): José Miguel Pérez, Soledad Le Clainche, José Manuel Vega〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉 〈p〉The objective of this work is to develop a procedure that allows for reconstructing three-dimensional flow fields from two-dimensional information contained in some representative planes, conveniently distributed throughout the domain. The reconstructing tool is based on two recent methods developed by two of the authors, namely the higher order dynamic mode decomposition and the spatio-temporal Koopman decomposition (STKD). The latter method decomposes a given spatio-temporal flow field as a series expansion in Fourier-like modes (including both wavenumbers/frequencies and spatial/temporal growth rates) in time and some distinguished longitudinal spatial coordinates, and spatial modes depending on the remaining transverse spatial coordinates. To obtain the (unknown) three-dimensional reconstruction, the STKD method is first applied to the (known) two-dimensional data in the considered planes. This application of STKD yields both the wavenumbers/frequencies and the spatial/temporal growth rates appearing in the three-dimensional reconstruction. Imposing that the three-dimensional reconstruction coincides with the two-dimensional data in the given set of planes, a system of linear equations results that permits computing the various ingredients appearing in the three-dimensional STKD expansion.〈/p〉 〈p〉The performance of the method is tested in the three-dimensional wake of a circular cylinder at Reynolds number (based on the cylinder diameter and the incoming free-stream velocity) equal to 280. The resulting flow is highly non-linear and quasi-periodic, with two fundamental temporal frequencies, associated with the well-known modes A and B. The method can be applied using experimental or numerical data, allowing to identify the full three-dimensional flow field and its main characteristics from limited information.〈/p〉 〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Annamaria Mazzia, Massimiliano Ferronato, Pietro Teatini, Claudia Zoccarato〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The prediction of long-term dynamics of transitional environments, e.g., lagoon evolution, salt-marsh growth or river delta progradation, is an important issue to estimate the potential impacts of different scenarios on such vulnerable intertidal morphologies. The numerical simulation of the combined accretion and consolidation, i.e., the two main processes driving the dynamics of these environments, however, suffers from a significant geometric non-linearity, which may result in a pronounced grid distortion using standard grid-based discretization methods. The present work describes a novel numerical approach, based on the Virtual Element Method (VEM), for the long-term simulation of the vertical dynamics of transitional landforms. The VEM is a grid-based variational technique for the numerical discretization of Partial Differential Equations (PDEs) allowing for the use of very irregular meshes consisting of a free combination of different polyhedral elements. The model solves the groundwater flow equation, coupled to a geomechanical module based on Terzaghi's principle, in a large-deformation setting, taking into account both the geometric and the material non-linearity. The use of the VEM allows for a great flexibility in the element generation and management, avoiding the numerical issues connected with the adoption of strongly distorted meshes. The numerical model is developed, implemented and tested in real-world examples, showing an interesting potential for addressing complex environmental situations.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Mustapha Ghilani, El Houssaine Quenjel, Mazen Saad〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉We are concerned with the approximation of solutions to a compressible two-phase flow model in porous media thanks to an enhanced control volume finite element discretization. The originality of the methodology consists in treating the case where the densities are depending on their own pressures without any major restriction neither on the permeability tensor nor on the mesh. Contrary to the ideas of [23], the point of the current scheme relies on a phase-by-phase “sub”-unpwinding approach so that we can recover the coercivity-like property. It allows on a second place for the preservation of the physical bounds on the discrete saturation. The convergence of the numerical scheme is therefore performed using classical compactness arguments. Numerical experiments are presented to exhibit the efficiency and illustrate the qualitative behavior of the implemented method.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Hannah Lu, Daniel M. Tartakovsky〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉Proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) are two complementary singular-value decomposition (SVD) techniques that are widely used to construct reduced-order models (ROMs) in a variety of fields of science and engineering. Despite their popularity, both DMD and POD struggle to formulate accurate ROMs for advection-dominated problems because of the nature of SVD-based methods. We investigate this shortcoming of conventional POD and DMD methods formulated within the Eulerian framework. Then we propose a Lagrangian-based DMD method to overcome this so-called translational problem. Our approach is consistent with the spirit of physics-aware DMD since it accounts for the evolution of characteristic lines. Several numerical tests are presented to demonstrate the accuracy and efficiency of the proposed Lagrangian DMD method.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Tao Li, Cheng Wang, Tonghui Yang, Dongping Chen, S. Chung Kim Yuen〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉The development of a novel Adaptive Mesh Enlargement (AME) method with ordinary Weight Essentially Non-Oscillatory (WENO) scheme for large-scale explosion problems is presented. The novel AME method adaptively adjusts the size of computational domains to accommodate the development of an explosion event which initially starts in a small domain. The grid resolution is verified by solving a simple one-dimensional (1-D) partial differential equation. The accuracy of AME methods is evaluated by calculations of 1-D Burgers equations and shock tube tests. It is found that the accuracy of AME methods depends on the number of enlargement operations. Numerical solutions with a reasonable number of enlargements are consistent with those of the fixed mesh method. At the same time, AME methods with multiple enlargements reduce the computational cost by several orders of magnitude. A three-dimensional (3-D) symmetric model with the AME method operating five enlargements is further developed to simulate the large-scale near-field explosion and validated against experiments. Propagation of the rarefaction and shock waves as well as the reflected wave are analyzed. The predicted peak incident pressures yield a variation of ∼3% compared to experiments indicating that the AME method is an effective construction method of computational domains for the problem of large-scale fluid dynamics.〈/p〉〈/div〉

Beschreibung: 〈p〉Publication date: 15 April 2020〈/p〉 〈p〉〈b〉Source:〈/b〉 Journal of Computational Physics, Volume 407〈/p〉 〈p〉Author(s): Dmitri Kuzmin, Nikita Klyushnev〈/p〉 〈h5〉Abstract〈/h5〉 〈div〉〈p〉This work introduces a new type of constrained algebraic stabilization for continuous piecewise-linear finite element approximations to the equations of ideal magnetohydrodynamics (MHD). At the first step of the proposed flux-corrected transport (FCT) algorithm, the Galerkin element matrices are modified by adding graph viscosity proportional to the fastest characteristic wave speed. At the second step, limited antidiffusive corrections are applied and divergence cleaning is performed for the magnetic field. The limiting procedure developed for this stage is designed to enforce local maximum principles, as well as positivity preservation for the density and thermodynamic pressure. Additionally, it adjusts the magnetic field in a way which penalizes divergence errors without violating conservation laws or positivity constraints. Numerical studies for 2D test problems are performed to demonstrate the ability of the proposed algorithms to accomplish this task in applications to ideal MHD benchmarks.〈/p〉〈/div〉