# Application of an Implicit Scheme of the Discontinuous Galerkin Method to Solving Gas Dynamics Problems on Nvidia Graphic Accelerators

V.F. Masyagin, R.V. Zhalnin, V.F. TishkinThe paper proposes an implicit scheme of the discontinuous Galerkin method for solving the gas dynamics equations on unstructured grids. The implicit scheme is based on the representation of the system of grid equations in "delta form". To solve the SLAE obtained during the approximation of the initial equations, solvers from the NVIDIA AmgX library are used. To verify the numerical algorithm, we calculate the flow of an inviscid compressible gas in a flat channel with a wedge and solve the problem of a viscous gas flow around a symmetric airfoil NACA0012. The results obtained are compared with the experimental results and the known numerical solutions to the presented problems. We conclude that the numerical and experimental data are in good agreement.Full text

- Keywords
- gas dynamics equations; discontinuous Galerkin method; implicit scheme; NVIDIA AmgX.
- References
- 1. Wei Su, Peng Wang, Yonghao Zhang. High-Order Hybridisable Discontinuous Galerkin Method for the Gas Kinetic Equation. International Journal of Computational Fluid Dynamics, 2019, vol. 33, no. 8, pp. 335-342. DOI: 10.1080/10618562.2019.1666110

2. Yunzhang Li, Chi-Wang Shu, Shanjian Tang. A Discontinuous Galerkin Method for Stochastic Conservation Laws. SIAM Journal on Scientific Computing, 2020, vol. 42, no. 1, pp. 54-86. DOI: 10.1137/19M125710X

3. Rhebergen S., Wells G. N. A Hybridizable Discontinuous Galerkin Method for the Navier-Stokes Equations with Pointwise Divergence-Free Velocity Field. Journal of Scientific Computing, 2018, no. 76, pp. 1484-1501. DOI: 10.1007/s10915-018-0671-4

4. Hajihassanpour M., Hejranfar K. A High-Order Nodal Discontinuous Galerkin Method to Solve Preconditioned Multiphase Euler/Navier-Stokes Equations for Inviscid/Viscous Cavitating Flows. International Journal for Numerical Methods in Fluids, 2020, vol. 92, no. 5, pp. 478-508. DOI: 10.1002/fld.4792

5. Ladonkina M.E., Nekliudova O.A., Ostapenko V.V., Tishkin V.F. [On Increasing the Stability of the Combined Scheme of the Discontinuous Galerkin Method]. Mathematical Models and Computer Simulations, 2021, vol. 33, no. 3, pp. 98-108. DOI: 10.20948/mm-2021-03-07. (in Russian)

6. Schall E., Chauchat N. Implicit Method and Slope Limiter in AHMR Procedure for High Order Discontinuous Galerkin Methods for Compressible Flows. Communications in Nonlinear Science and Numerical Simulation, 2019, vol. 72, pp. 371-391. DOI: 10.1016/j.cnsns.2018.12.020

7. Asada H., Kawai S. A Simple Cellwise High-order Implicit Discontinuous Galerkin Scheme for Unsteady Turbulent Flows. Transactions of the Japan Society for Aeronautical and Space Sciences, 2019, vol. 62, no. 2, pp. 93-107. DOI: 10.2322/tjsass.62.93

8. Luo H., Segawa H., Visbal M.R. An Implicit Discontinuous Galerkin Method for the Unsteady Compressible Navier-Stokes Equations. Computers & Fluids, 2012, vol. 50, pp. 133-144. DOI: 10.1016/j.compfluid.2011.10.009

9. Guthrey P.T., Rossmanith J.A. The Regionally Implicit Discontinuous Galerkin Method: Improving the Stability of DG-FEM. SIAM Journal on Numerical Analysis, 2019, vol. 57, no. 3, pp. 1263-1288. DOI: 10.1137/17M1156174

10. Volkov A.V. [Application of the Multigrid Approach for Solving 3D Navier-Stokes Equations on Hexahedral Grids Using the Discontinuous Galerkin Method]. Computational Mathematics and Mathematical Physics, 2010, vol. 50, no. 3, pp. 517-531. (in Russian) DOI: 10.1134/S0965542510030103.

11. Haijin Wang, Qiang Zhang, Shiping Wang, Chi-Wang Shu. Local Discontinuous Galerkin Methods with Explicit-Implicit-Null Time Discretizations for Solving Nonlinear Diffusion Problems. Science China Mathematics, 2020, vol. 63, pp. 183-204. DOI: 10.1007/s11425-018-9524-x

12. Haijin Wang, Qiang Zhang, Chi-Wang Shu. Third Order Implicit-Explicit Runge-Kutta Local Discontinuous Galerkin Methods with Suitable Boundary Treatment for Convection-Diffusion Problems with Dirichlet Boundary Conditions. Journal of Computational and Applied Mathematics, 2018, vol. 342, pp. 164-179. DOI: 10.1016/j.cam.2018.04.004

13. Haijin Wang, Qiang Zhang, Chi-Wang Shu. Implicit-Explicit Local Discontinuous Galerkin Methods with Generalized Alternating Numerical Fluxes for Convection-Diffusion Problems. Journal of Scientific Computing, 2019, vol. 81, pp. 2080-2114. DOI: 10.1007/s10915-019-01072-4

14. Bogdanov P.B., Gorobec A.V., Sukov S.A. [Adaptation and Optimization of Basic Operations for an Unstructured Mesh CFD Algorithm for Computation on Massively Parallel Accelerators]. Computational Mathematics and Mathematical Physics, 2013, vol. 53, no. 8, pp. 1360-1373. DOI: 10.7868/S0044466913080048

15. Simoncini V., Szvld D.B. Flexible Inner-Outer Krylov Subspace Methods. SIAM Journal on Numerical Analysis, 2019, vol. 40, no. 6, pp. 2219-2239. DOI: 10.1137/S0036142902401074

16. Naumov M., Arsaev M., Castonguay P. et al. AmgX: a Library for GPU Accelerated Algebraic Multigrid and Preconditioned Iterative Methods. SIAM Journal on Scientific Computing, 2015, vol. 37, no. 5, pp. 602-626. DOI: 10.1137/140980260

17. Zhalnin R.V., Maksimkin A.V., Masjagin V.F. et al. [Research of the Order of Accuracy of an Implicit Discontinuous Galerkin Method for Solving Problems of Gas Dynamics]. Middle Volga Mathematical Society Journal, 2015, vol. 17, no. 1, pp. 48-54. (in Russian)

18. Zhalnin R.V., Maksimkin A.V., Masjagin V.F. et al. [About the Use of WENO-Limiter in the Implicit Scheme for the Discontinuous Galerkin Method]. Middle Volga Mathematical Society Journal, 2015, vol. 17, no. 3, pp. 75-81. (in Russian)

19. Cockburn B., Shu C.-W. The Local Discontinuous Galerkin Finite Element Method for Convection-Diffusion Systems. SIAM Journal on Numerical Analysis, 1998, vol. 35, no. 6, pp. 2440-2463. DOI: 10.1137/S0036142997316712

20. Bassi F.A., Rebay S. A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations. Journal of Computational Physics, 1997, vol. 131, no. 2, pp. 267-279. DOI: 10.1006/jcph.1996.5572

21. Godunov S.K. [A Difference Method for Numerical Calculation of Discontinuous Solutions of the Equations of Hydrodynamics]. Matematicheskii Sbornik, 1959, vol. 47, no. 3, pp. 271-306. (in Russian)

22. Ladonkina M.E., Neklyudova O.A., Tishkin V.F. [Research of the Impact of Different Limiting Functions on the Order of Solution Obtained by RKDG]. Modern Problems of Science and Education, 2012, no. 034, available at: http://www.mathnet.ru/links/e9e6bfb5cb374bbfe8bb183e382941cc/ipmp52.pdf (in Russian)

23. Ladonkina M.E., Neklyudova O.A., Tishkin V.F. [Research of the Impact of Different Limiting Functions on the Order of Solution Obtained by RKDG]. Mathematical Models and Computer Simulations, 2012, vol. 24, no. 12, pp. 124-128. (in Russian)

24. Zhalnin R.V., Veselova E.A., Derjugin Ju.N. et al. [Software Package LOGOS. The High Order of Accuracy Method on Block-Structured Meshes with WENO Reconstruction]. Modern Problems of Science and Education, 2012, no. 6, 9 p, available at: http://science-education.ru/ru/article/view?id=7329 (in Russian)

25. Veselova E.A., Zhalnin R.V., Derjugin Ju.N. et al. [The Software LOGOS. Calculation Method for Viscous Compressible Gas Flows on a Block-Structured Meshes]. Modern Problems of Science and Education, 2014, no. 2, 22 p, available at: http://science-education.ru/ru/article/view?id=12601 (in Russian)

26. Volkov K.N., Derjugin Ju.N., Emel'janov V.N. [Implementation of Parallel Calculations on Graphics Processor Units in the LOGOS Computational Fluid Dynamics Package]. Numerical Methods and Programming, 2013, vol. 14, no. 3, pp. 334-342. (in Russian)

27. Harris C.D. Two-Dimensional Aerodynamic Characteristics of the NACA0012 Airfoil in the Langley 8-Foot Transonic Pressure Tunnel. NACA Technical Memorandum 81927. Langley Research Center, 1981.