Very high-order asymptotic-preserving schemes for hyperbolic systems of conservation laws with parabolic degeneracy on unstructured meshes. (English) Zbl 07325133

Summary: In this paper, we consider the numerical approximation of hyperbolic systems of conservation laws with stiff source terms and parabolic degeneracy in the asymptotic limit. We are more precisely interested in the design of high-order asymptotic-preserving schemes on unstructured meshes. Our approach is based on a very simple modification of the numerical flux associated with the usual HLL scheme and boils down to a sharp control of the underlying numerical diffusion. The strategy allows to capture the correct asymptotic parabolic behaviour and to preserve the high-order accuracy also in the asymptotic limit. Numerical experiments are proposed to illustrate these properties.


65-XX Numerical analysis
35-XX Partial differential equations


Full Text: DOI


[1] Berthon, C.; LeFloch, P. G.; Turpault, R., Late-time/stiff-relaxation asymptotic-preserving approximations of hyperbolic equations, Math. Comp., 82, 282, 831-860 (2013) · Zbl 1317.65182
[2] Jin, S., Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations, SIAM J. Sci. Comput., 21, 2, 441-454 (1999), (electronic) · Zbl 0947.82008
[3] Gosse, L.; Toscani, G., An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations, C. R. Math. Acad. Sci. Paris, 334, 4, 337-342 (2002) · Zbl 0996.65093
[4] Gosse, L., (Computing Qualitatively Correct Approximations of Balance Laws. Computing Qualitatively Correct Approximations of Balance Laws, SIMAI Springer Series, vol. 2 (2013), Springer: Springer Milan), Exponential-fit, well-balanced and asymptotic-preserving, xx+340 · Zbl 1272.65065
[5] Berthon, C.; Turpault, R., Asymptotic preserving HLL schemes, Numer. Methods Partial Differential Equations, 27, 6, 1396-1422 (2011) · Zbl 1237.65100
[6] Chalons, C.; Girardin, M.; Kokh, S., Large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms, SIAM J. Sci. Comput., 35, 6, A2874-A2902 (2013) · Zbl 1284.35262
[7] Blachère, F.; Turpault, R., An admissibility and asymptotic-preserving scheme for systems of conservation laws with source term on 2D unstructured meshes, J. Comput. Phys., 315, 98-123 (2016) · Zbl 1349.76308
[8] Berthon, C.; Moebs, G.; Sarazin-Desbois, C.; Turpault, R., An asymptotic-preserving scheme for systems of conservation laws with source terms on 2D unstructured meshes, Commun. Appl. Math. Comput. Sci., 11, 1, 55-77 (2016) · Zbl 1382.65266
[9] Blachère, F.; Turpault, R., An admissibility and asymptotic preserving scheme for systems of conservation laws with source term on 2D unstructured meshes with high-order MOOD reconstruction, Comput. Methods Appl. Mech. Engrg., 317, 836-867 (2017) · Zbl 1439.65102
[10] Chalons, C.; Turpault, R., High-order asymptotic-preserving schemes for linear systems: Application to the Goldstein-Taylor equations, Numer. Methods Partial Differential Equations, 35, 1538-1561 (2019) · Zbl 1418.65112
[11] Filbet, F.; Jin, S., A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys., 229, 20, 7625-7648 (2010) · Zbl 1202.82066
[12] Lafitte, P.; Lejon, A.; Samaey, G., A high-order asymptotic-preserving scheme for kinetic equations using projective integration, SIAM J. Numer. Anal., 54, 1, 1-33 (2016) · Zbl 1336.65147
[13] Boscarino, S.; LeFloch, P. G.; Russo, G., High-order asymptotic-preserving methods for fully nonlinear relaxation problems, SIAM J. Sci. Comput., 36, 2, A377-A395 (2014)
[14] Boscarino, S.; Pareschi, L.; Russo, G., Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit, SIAM J. Sci. Comput., 35, 1, A22-A51 (2013) · Zbl 1264.65150
[15] Harten, A.; Lax, P. D.; van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 1, 35-61 (1983) · Zbl 0565.65051
[16] Bianchini, S.; Hanouzet, B.; Natalini, R., Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Comm. Pure Appl. Math., 60, 11, 1559-1622 (2007) · Zbl 1152.35009
[17] Diot, S., La méthode MOOD Multi-dimensional Optimal Order Detection: la première approche a posteriori aux méthodes volumes finis d’ordre très élevé (2012), Université de Toulouse, Université Toulouse III-Paul Sabatier, August
[18] Spiteri, R. J.; Ruuth, S. J., A new class of optimal high-order strong-stability-preserving time discretization methods, SIAM J. Numer. Anal., 40, 2, 469-491 (2002), (electronic) · Zbl 1020.65064
[19] Diot, S.; Clain, S.; Loubère, R., Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials, Comput. Fluids, 64, 43-63 (2012) · Zbl 1365.76149
[20] Clain, S.; Figueiredo, J., The MOOD method for the non-conservative shallow-water system, Comput. Fluids, 145, 99-128 (2017) · Zbl 1390.76415
[21] Geuzaine, C.; Remacle, J.-F., Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg., 79, 11, 1309-1331 (2009) · Zbl 1176.74181
[22] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics (2009), Springer-Verlag: Springer-Verlag Berlin, A practical introduction, xxiv+724 · Zbl 1227.76006
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.