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  • 1
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    Nonlinear dynamics and numerical uncertainties in CFD (1996)
    In:  CASI
    Details
    Publication Date: 2019-06-28
    Description: The application of nonlinear dynamics to improve the understanding of numerical uncertainties in computational fluid dynamics (CFD) is reviewed. Elementary examples in the use of dynamics to explain the nonlinear phenomena and spurious behavior that occur in numerics are given. The role of dynamics in the understanding of long time behavior of numerical integrations and the nonlinear stability, convergence, and reliability of using time-marching, approaches for obtaining steady-state numerical solutions in CFD is explained. The study is complemented with spurious behavior observed in CFD computations.
    Keywords: Numerical Analysis
    Type: NASA-TM-110398 , NAS 1.15:110398 , A-96173
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  • 2
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    Dynamical Approach Study of Spurious Numerics in Nonlinear Computations (2002)
    In:  Other Sources
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    Publication Date: 2019-07-18
    Description: The last two decades have been an era when computation is ahead of analysis and when very large scale practical computations are increasingly used in poorly understood multiscale complex nonlinear physical problems and non-traditional fields. Ensuring a higher level of confidence in the predictability and reliability (PAR) of these numerical simulations could play a major role in furthering the design, understanding, affordability and safety of our next generation air and space transportation systems, and systems for planetary and atmospheric sciences, and in understanding the evolution and origin of life. The need to guarantee PAR becomes acute when computations offer the ONLY way of solving these types of data limited problems. Employing theory from nonlinear dynamical systems, some building blocks to ensure a higher level of confidence in PAR of numerical simulations have been revealed by the author and world expert collaborators in relevant fields. Five building blocks with supporting numerical examples were discussed. The next step is to utilize knowledge gained by including nonlinear dynamics, bifurcation and chaos theories as an integral part of the numerical process. The third step is to design integrated criteria for reliable and accurate algorithms that cater to the different multiscale nonlinear physics. This includes but is not limited to the construction of appropriate adaptive spatial and temporal discretizations that are suitable for the underlying governing equations. In addition, a multiresolution wavelets approach for adaptive numerical dissipation/filter controls for high speed turbulence, acoustics and combustion simulations will be sought. These steps are corner stones for guarding against spurious numerical solutions that are solutions of the discretized counterparts but are not solutions of the underlying governing equations.
    Keywords: Numerical Analysis
    Type: Short Course in Numerical Methods in Mechanics; Feb 10, 2003 - Feb 13, 2003; Tullahoma, TN; United States
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  • 3
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    Extension of Efficient Low Dissipative High Order Schemes for 3-D Curvilinear Moving Grids (2000)
    In:  Other Sources
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    Publication Date: 2019-07-17
    Description: The efficient low dissipative high order schemes proposed by Yee et al. is formulated for 3-D curvilinear moving grids. These schemes consists of a high order base schemes combined with nonlinear characteristic filters. The amount of numerical dissipation is minimized by applying the schemes to the entropy splitting form of the inviscid flux derivatives. The analysis is given for a thermally perfect gas. The main difficulty in the extension of higher order schemes that were formulated in Cartesian coordinates to curvilinear moving grids is the higher order transformed metric evaluations. The higher order numerical evaluation of the transformed metric terms to insure freestream preservation is done in a coordinate invariant manner. The formulation is an improvement over existing formulation of high order scheme in curvilinear moving grids.
    Keywords: Numerical Analysis
    Type: Computing the Future III: Frontiers of CFD 2000 Symposium; Jun 26, 2000 - Jun 28, 2000; Half Moon Bay, CA; United States
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  • 4
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    Performance of Several High Order Numerical Methods for Supersonic Combustion (2001)
    In:  Other Sources
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    Publication Date: 2019-07-18
    Description: The performance of two recently developed numerical methods by Yee et al. and Sjoegreen and Yee using postprocessing nonlinear filters is examined for a 2-D multiscale viscous supersonic react-live flow. These nonlinear filters can improve nonlinear instabilities and at the same time can capture shock/shear waves accurately. They do not, belong to the class of TVD, ENO or WENO schemes. Nevertheless, they combine stable behavior at discontinuities and detonation without smearing the smooth parts of the flow field. For the present study, we employ a fourth-order Runge-Kutta in time and a sixth-order non-dissipative spatial base scheme for the convection and viscous terms. We denote the resulting nonlinear filter schemes ACM466-RK4 and WAV66-RK4.
    Keywords: Numerical Analysis
    Type: 9th International Conference on Numerical Combustion; Apr 07, 2002 - Apr 10, 2002; Sorrento; Italy
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  • 5
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    High Order Finite Difference Methods with Subcell Resolution for 2D Detonation Waves (2012)
    In:  CASI
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    Publication Date: 2019-07-13
    Description: In simulating hyperbolic conservation laws in conjunction with an inhomogeneous stiff source term, if the solution is discontinuous, spurious numerical results may be produced due to different time scales of the transport part and the source term. This numerical issue often arises in combustion and high speed chemical reacting flows.
    Keywords: Numerical Analysis
    Type: ARC-E-DAA-TN4507 , Center for Turbulence Research Annual Research Brief; Dec 01, 2011; Stanford, CA; United States
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  • 6
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    On the Importance of the Dynamics of Discretizations (1995)
    In:  Other Sources
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    Publication Date: 2019-07-18
    Description: It has been realized recently that the discrete maps resulting from numerical discretizations of differential equations can possess asymptotic dynamical behavior quite different from that of the original systems. This is the case not only for systems of Ordinary Differential Equations (ODEs) but in a more complicated manner for Partial Differential Equations (PDEs) used to model complex physics. The impact of the modified dynamics may be mild and even not observed for some numerical methods. For other classes of discretizations the impact may be pronounced, but not always obvious depending on the nonlinear model equations, the time steps, the grid spacings and the initial conditions. Non-convergence or convergence to periodic solutions might be easily recognizable but convergence to incorrect but plausible solutions may not be so obvious - even for discretized parameters within the linearized stability constraint. Based on our past four years of research, we will illustrate some of the pathology of the dynamics of discretizations, its possible impact and the usage of these schemes for model nonlinear ODEs, convection-diffusion equations and grid adaptations.
    Keywords: Numerical Analysis
    Type: 3rd International Congress on Industrial and Applied Mathematics; Jul 03, 1995 - Jul 07, 1995; Hamburg; Germany
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  • 7
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    High Order Filter Methods for the Non-ideal Compressible MHD Equations (2003)
    In:  Other Sources
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    Publication Date: 2019-07-18
    Description: The generalization of a class of low-dissipative high order filter finite difference methods for long time wave propagation of shock/turbulence/combustion compressible viscous gas dynamic flows to compressible MHD equations for structured curvilinear grids has been achieved. The new scheme is shown to provide a natural and efficient way for the minimization of the divergence of the magnetic field numerical error. Standard divergence cleaning is not required by the present filter approach. For certain non-ideal MHD test cases, divergence free preservation of the magnetic fields has been achieved.
    Keywords: Numerical Analysis
    Type: 10th International Conference on Hyperbolic Problems: Theory, Numerics, Applications (HYP2004); Sep 13, 2004 - Sep 17, 2004; Osaka; Japan
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  • 8
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    A Numerical Study of Resistivity and Hall Effects for a Compressible MHD Model (2005)
    In:  CASI
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    Publication Date: 2019-07-13
    Description: The effect of resistive, Hall, and viscous terms on the flow structure compared with compressible ideal MHD is studied numerically for a one-fluid non-ideal MHD model. The goal of the present study is to shed some light on the emerging area of non-ideal MHD modeling and simulation. Numerical experiments are performed on a hypersonic blunt body flow with future application to plasma aerodynamics flow control in reentry vehicles. Numerical experiments are also performed on a magnetized time-developing mixing layer with possible application to magnetic/turbulence mixing.
    Keywords: Numerical Analysis
    Type: 36th Annual AIAA Plasmadynamics and Lasers Conference; Jun 06, 2005 - Jun 09, 2005; Toronto, Ontario; Canada
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  • 9
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    Dynamical Approach Study of Spurious Steady-State Numerical Solutions of Nonlinear Differential Equations (1995)
    In:  CASI
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    Publication Date: 2019-07-13
    Description: The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how 'numerical' basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DEs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless of the stability of the spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.
    Keywords: Numerical Analysis
    Type: NASA-TM-112909 , NAS 1.15:112909 , RNR-92-008 , Numerical Methods in Fluid Mechanics; Sep 25, 1991 - Sep 27, 1991; Lausanne; Switzerland|Computer Fluid Dynamics; 4; 219-283
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  • 10
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    The Dynamics of Some Iterative Implicit Schemes (1994)
    In:  CASI
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    Publication Date: 2019-07-13
    Description: The global asymptotic nonlinear behavior of some standard iterative procedures in solving nonlinear systems of algebraic equations arising from four implicit linear multistep methods (LMMs) in discretizing 2 x 2 systems of first-order autonomous nonlinear ordinary differential equations is analyzed using the theory of dynamical systems. With the aid of parallel Connection Machines (CM-2 and CM-5), the associated bifurcation diagrams as a function of the time step, and the complex behavior of the associated 'numerical basins of attraction' of these iterative implicit schemes are revealed and compared. Studies showed that all of the four implicit LMMs exhibit a drastic distortion and segmentation but less shrinkage of the basin of attraction of the true solution than standard explicit methods. The numerical basins of attraction of a noniterative implicit procedure mimic more closely the basins of attraction of the differential equations than the iterative implicit procedures for the four implicit LMMs.
    Keywords: Numerical Analysis
    Type: NASA-TM-112670 , NAS 1.15:112670 , Contemporary Mathematics (ISSN 0271-4132); 172; 75-96
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