zbMATH — the first resource for mathematics

Adaptive mesh refinement on graphics processing units for applications in gas dynamics. (English) Zbl 1451.65139
Summary: We present novel algorithms for cell-based adaptive mesh refinement on unstructured meshes of triangles on graphics processing units. Our implementation makes use of improved memory management techniques and a coloring algorithm for avoiding race conditions. Both the solver and AMR algorithms are entirely implemented on the GPU, with negligible communication between device and host. We show that the overhead of the AMR subroutines is small compared to the high order solver and that the proportion of total runtime spent adaptively refining the mesh decreases with the order of approximation. We apply our code to a number of benchmark problems as well as more recently proposed problems for the Euler equations that require extremely high resolution. We present the solution to a shock reflection problem that addresses the von Neumann triple point paradox with an accurately computed triple point location. Finally, we present the first solution on the full Euler equations to the problem of shock disappearance and self-similar diffraction of weak shocks around thin films.
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76M10 Finite element methods applied to problems in fluid mechanics
35L65 Hyperbolic conservation laws
65Y20 Complexity and performance of numerical algorithms
76N15 Gas dynamics, general
65Y05 Parallel numerical computation
Full Text: DOI
[1] Karakus, A.; Warburton, T.; Aksel, M.; Sert, C., A GPU-accelerated adaptive discontinuous Galerkin method for level set equation, Int. J. Comput. Fluid Dyn., 30, 1, 56-68 (2016) · Zbl 1359.65203
[2] Chan, J.; Wang, Z.; Modave, A.; Remacle, J.-F.; Warburton, T., GPU-accelerated discontinuous Galerkin methods on hybrid meshes, J. Comput. Phys., 318, 142-168 (2016) · Zbl 1349.65443
[3] Fuhry, M.; Giuliani, A.; Krivodonova, L., Discontinuous Galerkin methods on graphics processing units for nonlinear hyperbolic conservation laws, Int. J. Numer. Methods Fluids, 76, 12, 982-1003 (2014)
[4] Gandham, R.; Medina, D.; Warburton, T., GPU accelerated discontinuous Galerkin methods for shallow water equations, Commun. Comput. Phys., 18, 1, 37-64 (2015) · Zbl 1373.76086
[5] Chan, J.; Warburton, T., GPU-accelerated Bernstein-Bézier discontinuous Galerkin methods for wave problems, SIAM J. Sci. Comput., 39, 2, A628-A654 (2017) · Zbl 1365.65219
[6] de la Asunción, M.; Castro, M., Simulation of tsunamis generated by landslides using adaptive mesh refinement on GPU, J. Comput. Phys., 345, 91-110 (2017)
[7] Abdi, D. S.; Wilcox, L. C.; Warburton, T. C.; Giraldo, F. X., A GPU-accelerated continuous and discontinuous Galerkin non-hydrostatic atmospheric model, Int. J. High Perform. Comput. Appl., Article 1094342017694427 pp. (2017)
[8] Giuliani, A.; Krivodonova, L., Face coloring in unstructured CFD codes, Parallel Comput., 63, 17-37 (2017)
[9] Burstedde, C.; Ghattas, O.; Gurnis, M.; Isaac, T.; Stadler, G.; Warburton, T.; Wilcox, L., Extreme-scale AMR, (Proceedings of the 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis (2010), IEEE Computer Society), 1-12
[10] Beckingsale, D.; Gaudin, W.; Herdman, A.; Jarvis, S., Resident block-structured adaptive mesh refinement on thousands of graphics processing units, (44th International Conference on Parallel Processing. 44th International Conference on Parallel Processing, ICPP (2015), IEEE), 61-70
[11] Pain, C.; Umpleby, A.; De Oliveira, C.; Goddard, A., Tetrahedral mesh optimisation and adaptivity for steady-state and transient finite element calculations, Comput. Methods Appl. Mech. Eng., 190, 29-30, 3771-3796 (2001) · Zbl 1008.76041
[12] Schnepp, S. M.; Weiland, T., Efficient large scale electromagnetic simulations using dynamically adapted meshes with the discontinuous Galerkin method, J. Comput. Appl. Math., 236, 18, 4909-4924 (2012) · Zbl 1458.78023
[13] Eskilsson, C., An hp-adaptive discontinuous Galerkin method for shallow water flows, Int. J. Numer. Methods Fluids, 67, 11, 1605-1623 (2011) · Zbl 1381.76165
[14] MacNeice, P.; Olson, K. M.; Mobarry, C.; De Fainchtein, R.; Packer, C., PARAMESH: a parallel adaptive mesh refinement community toolkit, Comput. Phys. Commun., 126, 3, 330-354 (2000) · Zbl 0953.65088
[15] Adams, M., CHOMBO Software Package for AMR Applications-Design Document (2011), Lawrence Berkeley National Laboratory technical report lbnl-6616e
[16] Bangerth, W.; Hartmann, R.; Kanschat, G., deal.II - a general purpose object oriented finite element library, ACM Trans. Math. Softw., 33, 4, 24/1-24/27 (2007) · Zbl 1365.65248
[17] Mandli, K. T.; Ahmadia, A. J.; Berger, M.; Calhoun, D.; George, D. L.; Hadjimichael, Y.; Ketcheson, D. I.; Lemoine, G. I.; LeVeque, R. J., Clawpack: building an open source ecosystem for solving hyperbolic PDEs, PeerJ Comput. Sci., 2, Article e68 pp. (2016)
[18] Berger, M. J.; Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations, J. Comput. Phys., 53, 3, 484-512 (1984) · Zbl 0536.65071
[19] Zheng, J. Z.X., Block-Based Adaptive Mesh Refinement Finite-Volume Scheme for Hybrid Multi-Block Meshes (2012), PhD thesis
[20] Ivan, L.; Sterck, H. D.; Northrup, S. A.; Groth, C., Multi-dimensional finite-volume scheme for hyperbolic conservation laws on three-dimensional solution-adaptive cubed-sphere grids, J. Comput. Phys., 255, 205-227 (2013) · Zbl 1349.76340
[21] Burstedde, C.; Wilcox, L. C.; Ghattas, O., p4est: scalable algorithms for parallel adaptive mesh refinement on forests of octrees, SIAM J. Sci. Comput., 33, 3, 1103-1133 (2011) · Zbl 1230.65106
[22] Devine, K. D.; Flaherty, J. E., Parallel adaptive hp-refinement techniques for conservation laws, Appl. Numer. Math., 20, 4, 367-386 (1996) · Zbl 0860.65095
[23] Richter, T., Parallel Multigrid Method for Adaptive Finite Elements with Application to 3D Flow Problems (2005), University of Heidelberg, PhD thesis
[24] Zanotti, O.; Dumbser, M., A high order special relativistic hydrodynamic and magnetohydrodynamic code with space-time adaptive mesh refinement, Comput. Phys. Commun., 188, 110-127 (2015) · Zbl 1344.76058
[25] Rokos, G.; Gorman, G.; Kelly, P. H., Accelerating anisotropic mesh adaptivity on nVIDIA’s CUDA using texture interpolation, (European Conference on Parallel Processing (2011), Springer), 387-398
[26] Xia, Y.; Lou, J.; Luo, H.; Edwards, J.; Mueller, F., OpenACC acceleration of an unstructured CFD solver based on a reconstructed discontinuous Galerkin method for compressible flows, Int. J. Numer. Methods Fluids, 78, 3, 123-139 (2015)
[27] Tesdall, A. M., High resolution solutions for the supersonic formation of shocks in transonic flow, J. Hyperbolic Differ. Equ., 8, 03, 485-506 (2011) · Zbl 1233.35143
[28] Tesdall, A. M.; Hunter, J. K., Self-similar solutions for the diffraction of weak shocks, J. Comput. Sci., 4, 1-2, 92-100 (2013)
[29] Dubiner, M., Spectral methods on triangles and other domains, J. Sci. Comput., 6, 345-390 (Dec 1991)
[30] Merrill, D., CUB, version 1.5.1
[31] Giuliani, A.; Krivodonova, L., Analysis of slope limiters on unstructured triangular meshes, J. Comput. Phys., 374, 1-26 (2018)
[32] Berger, M. J.; LeVeque, R. J., Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems, SIAM J. Numer. Anal., 35, 6, 2298-2316 (1998) · Zbl 0921.65070
[33] Schaal, K.; Bauer, A.; Chandrashekar, P.; Pakmor, R.; Klingenberg, C.; Springel, V., Astrophysical hydrodynamics with a high-order discontinuous Galerkin scheme and adaptive mesh refinement, Mon. Not. R. Astron. Soc., 453, 4, 4278-4300 (2015)
[34] Bangerth, W.; Rannacher, R., Adaptive Finite Element Methods for Differential Equations (2013), Birkhäuser · Zbl 0948.65098
[35] Vincent, P. E.; Castonguay, P.; Jameson, A., Insights from von Neumann analysis of high-order flux reconstruction schemes, J. Comput. Phys., 230, 22, 8134-8154 (2011) · Zbl 1343.65117
[36] Chalmers, N.; Krivodonova, L., A robust CFL condition for the discontinuous Galerkin method on triangular meshes, Draft, available at · Zbl 1372.65270
[37] Springel, V., E pur si muove: Galilean-invariant cosmological hydrodynamical simulations on a moving mesh, Mon. Not. R. Astron. Soc., 401, 2, 791-851 (2010)
[38] Woodward, P.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. Comput. Phys., 54, 1, 115-173 (1984) · Zbl 0573.76057
[39] Guderley, K. G., Considerations on the Structure of Mixed Subsonic-Supersonic Flow Patterns (1947), Headquarters Air Materiel Command
[40] Tesdall, A. M.; Sanders, R.; Keyfitz, B. L., Self-similar solutions for the triple point paradox in gasdynamics, SIAM J. Appl. Math., 68, 5, 1360-1377 (2008) · Zbl 1147.76036
[41] Vasilev, E. I.; Kraiko, A. N., Numerical simulation of weak shock diffraction over a wedge under the von Neumann paradox conditions, Zh. Vychisl. Mat. Mat. Fiz., 39, 8, 1393-1404 (1999)
[42] Skews, B. W.; Ashworth, J. T., The physical nature of weak shock wave reflection, J. Fluid Mech., 542, 105-114 (2005) · Zbl 1078.76505
[43] Zakharian, A.; Brio, M.; Hunter, J.; Webb, G., The von Neumann paradox in weak shock reflection, J. Fluid Mech., 422, 193-205 (2000) · Zbl 0995.76042
[44] Tesdall, A. M.; Sanders, R.; Popivanov, N., Further results on Guderley Mach reflection and the triple point paradox, J. Sci. Comput., 64, 3, 721-744 (2015) · Zbl 06499229
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.