Efficient Parallelization of a Shock Capturing for Discontinuous Galerkin Methods using Finite Volume Sub-cells

We present a shock capturing procedure for high order Discontinuous Galerkin methods, by which shock regions are refined in sub-cells and treated by finite volume techniques. Hence, our approach combines the good properties of the Discontinuous Galerkin method in smooth parts of the flow with the perfect properties of a total variation diminishing finite volume method for resolving shocks without spurious oscillations. Due to the sub-cell approach the interior resolution on the Discontinuous Galerkin grid cell is nearly preserved and the number of degrees of freedom remains the same. This structure allows the interpretation of the data either as DG solution or as finite volume solution on the subgrid. In this paper we explain the efficient implementation of this coupled method on massively parallel computers and show some numerical results.

[1]  K. P.,et al.  HIGH RESOLUTION SCHEMES USING FLUX LIMITERS FOR HYPERBOLIC CONSERVATION LAWS * , 2012 .

[2]  The Multicore Challenge: Petascale DNS of a Spatially-Developing Supersonic Turbulent Boundary Layer Up to High Reynolds Numbers Using DGSEM , 2015 .

[3]  Eitan Tadmor,et al.  Solution of two‐dimensional Riemann problems for gas dynamics without Riemann problem solvers , 2002 .

[4]  Chunlei Liang,et al.  Computation of flows with shocks using the Spectral Difference method with artificial viscosity, I: Basic formulation and application , 2014 .

[5]  A. Jameson,et al.  Numerical solution of the Euler equations by finite volume methods using Runge Kutta time stepping schemes , 1981 .

[6]  F. Nicoud,et al.  Large-Eddy Simulation of the Shock/Turbulence Interaction , 1999 .

[7]  Pierre Sagaut,et al.  A problem-independent limiter for high-order Runge—Kutta discontinuous Galerkin methods , 2001 .

[8]  Claus-Dieter Munz,et al.  High Fidelity Scale-Resolving Computational Fluid Dynamics Using the High Order Discontinuous Galerkin Spectral Element Method , 2016 .

[9]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[10]  Jianxian Qiu,et al.  Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case , 2004 .

[11]  Eleuterio F. Toro,et al.  Numerical Methods for Wave Propagation , 2011 .

[12]  R. Courant,et al.  Über die partiellen Differenzengleichungen der mathematischen Physik , 1928 .

[13]  Arne Taube,et al.  Shock detection and limiting strategies for high order discontinuous Galerkin schemes , 2009 .

[14]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[15]  P. Raviart,et al.  On a Finite Element Method for Solving the Neutron Transport Equation , 1974 .

[16]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[17]  Michael Dumbser,et al.  Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes - Speed comparisons with Runge-Kutta methods , 2013, J. Comput. Phys..

[18]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[19]  Chi-Wang Shu,et al.  Discontinuous Galerkin Methods: Theory, Computation and Applications , 2011 .

[20]  Bernardo Cockburn,et al.  The Runge-Kutta local projection P1-discontinuous-Galerkin finite element method for scalar conservation laws , 1988 .

[21]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[22]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[23]  Rosa Donat,et al.  Shock-Vortex Interactions at High Mach Numbers , 2003, J. Sci. Comput..

[24]  J. Tinsley Oden,et al.  A discontinuous hp finite element method for the Euler and Navier–Stokes equations , 1999 .

[25]  David L. Darmofal,et al.  Shock capturing with PDE-based artificial viscosity for DGFEM: Part I. Formulation , 2010, J. Comput. Phys..

[26]  J. Nitsche.  Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[27]  Michael Dumbser,et al.  A sub-cell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG+HWENO schemes , 2004, J. Comput. Phys..

[28]  David A. Kopriva,et al.  Implementing Spectral Methods for Partial Differential Equations , 2009 .

[29]  Philip L. Roe,et al.  Discrete models for the numerical analysis of time-dependent multidimensional gas dynamics , 1986 .

[30]  P. Roe.  CHARACTERISTIC-BASED SCHEMES FOR THE EULER EQUATIONS , 1986 .

[31]  Claus-Dieter Munz,et al.  Explicit Discontinuous Galerkin methods for unsteady problems , 2012 .

[32]  David A. Kopriva,et al.  Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers , 2009 .

[33]  J. Peraire,et al.  Sub-Cell Shock Capturing for Discontinuous Galerkin Methods , 2006 .

[34]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[35]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[36]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[37]  Michael Dumbser,et al.  A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws , 2014, J. Comput. Phys..

[38]  George Em Karniadakis,et al.  The Development of Discontinuous Galerkin Methods , 2000 .

[39]  Chi-Wang Shu.  Discontinuous Galerkin Methods , 2010 .

[40]  R. D. Richtmyer,et al.  A Method for the Numerical Calculation of Hydrodynamic Shocks , 1950 .

[41]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme , 1974 .

[42]  Comparison of Realizable Schemes for the Eulerian Simulation of Disperse Phase Flows , 2014 .

[43]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[44]  C. Schulz-Rinne,et al.  Classification of the Riemann problem for two-dimensional gas dynamics , 1991 .

[45]  Antonio Huerta,et al.  A simple shock‐capturing technique for high‐order discontinuous Galerkin methods , 2012 .