A space–time adaptive discontinuous Galerkin scheme
Introduction
An important advantage of the discontinuous Galerkin (DG) approach consists of its high flexibility with respect to the spatial approximation: The approximation of the solution is represented by a piecewise polynomial and the degree of the local polynomials determine the order of accuracy in space. Hence, increasing the order can be achieved through a simple enrichment of the local basis. This can also be handled locally in regions where we aim a higher resolution. Geometrical flexibility is also an important feature of the DG method since it can be formulated on general unstructured grids with non-conforming meshes to handle complex geometries. The DG method is thus an ideal candidate, for which general adaptive strategies as local grid refinement (h-refinement) or local choice of the order of accuracy (p-adaptivity) can be applied, see, e.g., [25]. Due to this distinguished versatile flexibility the DG method is particularly suited for multi-scale problems which require small regions of high resolution while other parts can be discretized uniformly.
Adaptive concepts for discontinuous Galerkin schemes have been successfully applied, e.g., by Burgess and Mavriplis [7], Kopera and Giraldo [26] or Hartmann and Houston [22], [23]. All these approaches based on DG schemes with implicit or semi-implicit time integration. As we are particularly interested in unsteady flow problems, the adaptivity of the time approximation is an additional issue and is a focal point in this paper. Especially for wave dominated problems like acoustics and propagating shock waves, the time accuracy has to be preserved. Regions with local refinement and unsteady phenomena need small time steps. Allowing large time steps for a transient problem would lead to numerical diffusion and to loss of information. For an explicit scheme, a global time-stepping approach can cause inefficiency, since the smallest time step forced by small sized grid cells dominates the whole simulation. To overcome this inefficiency, we employ an explicit time approximation with local time-stepping, which allows each grid cell to take the maximum locally stable time step. The local time-stepping enables a powerful adaptation framework, which also takes the coupling of spatial and temporal scales into account since any local h-refinement or p-enrichment in space inherently implies an adaptation of the time step as well. Together with the explicit local time-stepping approach, the artificial viscosity based shock capturing introduced by Persson and Peraire for an implicit DG scheme [35] now also becomes an attractive method to robustly resolve shock waves.
In the following we describe an explicit discontinuous Galerkin scheme that incorporates the following properties:
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high order accurate spatial approximations on general unstructured grids with triangles, quadrilaterals, tetrahedra, hexahedra, pyramids, prisms,
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arbitrary high order in space and time,
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non-conforming meshes in 2 and 3 dimensions,
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fully conservative,
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shock capturing with artificial viscosity,
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local time-stepping with the time step from the local stability constraint,
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-adaptivity in 2 dimensions and p-adaptivity in 3 dimensions.
Different building blocks of our DG approach have been constructed and already been published in a series of papers [30], [15], [16], [19]. The scope of this work is the investigation of the combination and interaction of explicit local time-stepping, artificial viscosity based shock capturing and -adaptivity. To demonstrate the high potential of the adaptivity framework, simulation results of various complex unsteady flow problems are presented.
The paper is organized as follows. We first describe the main components of the space approximation within the semi-discrete approach in Section 2. The time consistent local time-stepping method and the determination of the time step are introduced in Section 3 to obtain the fully discrete formulation. The adaptivity framework (h- and p-adaptivity) and its implementation with respect to the local time-stepping scheme is discussed in Section 4. In Section 4 we also provide a brief overview of the artificial viscosity based shock capturing as part of the whole adaptation strategy. In Section 5, we focus on the simulation results and present different applications of the space–time-adaptive framework for two- and three-dimensional complex flows and also address the parallel performance and dynamic load balancing attributes of the implementation. Conclusions and final remarks are given in Section 6.
Section snippets
Approximation in space
For simplicity we restrict the derivations to a scalar advection–diffusion equation, which is formulated in conservation form asHere, denotes the conserved solution variable, is the advection flux and the diffusion flux. The diffusion in the flux formulation may be rewritten aswhere is the non-linear diffusion coefficient.
First, we consider the discretization in space. The spatial domain Ω is subdivided into
Time-consistent local time-stepping
The locality of the proposed explicit space–time discontinuous Galerkin scheme allows a completely new time-marching technique: Each grid cell may run with its own time step in a time-consistent manner. The local time-stepping algorithm is explained in the next subsection, followed by the determination of the maximum time step in each element.
Adaptivity in the space–time domain
The discontinuous trial space allows spatial adaptivity by changing the mesh size (h-adaptivity) or changing the local polynomial degree (p-adaptivity) in a straightforward way. The strategy to extend this to the space–time domain with respect to the local time-stepping is described in the following, whereas h-adaptivity is applied in two dimensions and p-adaptivity in two and three dimensions.
Numerical results
In this section, we first investigate the two-dimensional setup with shock capturing. The -adaptivity with local time stepping is demonstrated on a transonic laminar flow around an airfoil. The combination of local time stepping and the capturing of strong moving shocks is presented for the standard double Mach reflection problem [45]. In three-dimensions, we combine the local time stepping with p-adaptivity to simulate the flow past a sphere. Finally, we present a simulation of an unsteady
Conclusions
We described a high order discontinuous Galerkin scheme for the numerical solution of unsteady convection–diffusion equations. The DG scheme is formulated in a space–time domain based on a predictor–corrector formulation and generalizes the solution in the small approach of the finite volume framework. The space–time discretization is kept explicit by starting with the predictor that determines approximate values at all intermediate time levels needed for high order accuracy in time. The
Acknowledgements
The research presented in this paper was supported in parts by the Deutsche Forschungsgemeinschaft (DFG) in the context of the Cluster of Excellence Simulation Technology (SimTech), the Research Training Group GRK 1095 ‘Aero-thermodynamic design of a scramjet propulsion system’ and the German Ministry of Research (BMBF) in the collaborative project Software for Scalable Parallel Computers.
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