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Adaptive mesh refinement for hyperbolic systems based on third-order compact WENO reconstruction. (English) Zbl 1335.65077
Summary: We generalise to non-uniform grids of quad-tree type the compact weighted essentially non-oscillatory (WENO) reconstruction of D. Levy et al. [SIAM J. Sci. Comput. 22, No. 2, 656–672 (2000; Zbl 0967.65089)], thus obtaining a truly two-dimensional non-oscillatory third order reconstruction with a very compact stencil and that does not involve mesh-dependent coefficients. This latter characteristic is quite valuable for its use in \(h\)-adaptive numerical schemes, since in such schemes the coefficients that depend on the disposition and sizes of the neighbouring cells (and that are present in many existing WENO-like reconstructions) would need to be recomputed after every mesh adaption. In the second part of the paper we propose a third order \(h\)-adaptive scheme with the above-mentioned reconstruction, an explicit third order TVD Runge-Kutta scheme and the entropy production error indicator proposed by G. Puppo and M. Semplice [“Numerical entropy and adaptivity for finite volume schemes”, Commun. Comput. Phys. 10, No. 5, 1132–1160 (2011)]. After devising some heuristics on the choice of the parameters controlling the mesh adaption, we demonstrate with many numerical tests that the scheme can compute numerical solution whose error decays as \(\langle N\rangle^{-3}\), where \(\langle N\rangle\) is the average number of cells used during the computation, even in the presence of shock waves, by making a very effective use of \(h\)-adaptivity and the proposed third order reconstruction.

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35L65 Hyperbolic conservation laws
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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[1] Abgrall, R, On essentially non-oscillatory schemes on unstructured meshes: analysis and implementation, J. Comput. Phys., 114, 45-58, (1994) · Zbl 0822.65062
[2] Aràndiga, F; Baeza, A; Belda, AM; Mulet, P, Analysis of WENO schemes for full and global accuracy, SIAM J. Numer. Anal., 49, 893-915, (2011) · Zbl 1233.65051
[3] Arvanitis, C; Delis, AI, Behavior of finite volume schemes for hyperbolic conservation laws on adaptive redistributed spatial grids, SIAM J. Sci. Comput., 28, 1927-1956, (2006) · Zbl 1213.35300
[4] Bastian, P., Blatt, M., Dedner, A., Engwer, C., Fahlke, J., Gräser, C., Klöfkorn, R., Nolte, M., Ohlberger, M., Sander, O.: DUNE web page (2011). http://www.dune-project.org · Zbl 1373.76140
[5] Bastian, P; Blatt, M; Dedner, A; Engwer, C; Klöfkorn, R; Kornhuber, R; Ohlberger, M; Sander, O, A generic grid interface for parallel and adaptive scientific computing. part II: implementation and tests in DUNE, Computing, 82, 121-138, (2008) · Zbl 1151.65088
[6] Bastian, P; Blatt, M; Dedner, A; Engwer, C; Klöfkorn, R; Ohlberger, M; Sander, O, A generic grid interface for parallel and adaptive scientific computing. part I: abstract framework, Computing, 82, 103-119, (2008) · Zbl 1151.65089
[7] Berger, MJ; LeVeque, RJ, Adaptive mesh refinement using wave-propagation algorithms for hyperbolic systems, SIAM J. Numer. Anal., 35, 2298-2316, (1998) · Zbl 0921.65070
[8] Burri, A., Dedner, A., Klöfkorn, R., Ohlberger, M.: An efficient implementation of an adaptive and parallel grid in dune. In: Notes on Numerical Fluid Mechanics, vol. 91, pp. 67-82 (2006). See also http://aam.mathematik.uni-freiburg.de/IAM/Research/alugrid · Zbl 0967.65089
[9] Čada, M; Torrilhon, M, Compact third-order limiter functions for finite volume methods, J. Comput. Phys., 228, 4118-4145, (2009) · Zbl 1273.76286
[10] Constantinescu, EM; Sandu, A, Multirate timestepping methods for hyperbolic conservation laws, J. Sci. Comput., 33, 239-278, (2007) · Zbl 1127.76033
[11] Cravero, I., Semplice, M.: On the accuracy of WENO and CWENO reconstructions of third order on nonuniform meshes. Preprint arXiv:1503.00736 · Zbl 1343.65116
[12] Davies-Jones, R, Comments on “kinematic analysis of frontogenesis associated with a nondivergent vortex”, J. Atmos. Sci., 42, 2073-2075, (1985)
[13] Dumbser, M; Käser, M, Arbitrary high order non-oscillatory finite volume schemes on unstructured meshes for linear hyperbolic systems, J. Comput. Phys., 221, 693-723, (2007) · Zbl 1110.65077
[14] Feng, H; Hu, F; Wang, R, A new mapped weighted essentially non-oscillatory scheme, J. Sci. Comput., 51, 449-473, (2012) · Zbl 1253.65124
[15] Fürst, J, A weighted least square scheme for compressible flows, Flow Turbul. Combust., 76, 331-342, (2006) · Zbl 1123.76039
[16] Gottlieb, S., Ketcheson, D., Shu, C.W.: Strong Stability Preserving Runge-Kutta and Multistep Time Discretizations. World Scientific Publishing, Hackensack (2011) · Zbl 1241.65064
[17] Henrick, AK; Aslam, TD; Powers, JM, Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points, J. Comput. Phys., 207, 542-567, (2005) · Zbl 1072.65114
[18] Hu, C; Shu, CW, Weighted essentially non-oscillatory schemes on triangular meshes, J. Comput. Phys., 150, 97-127, (1999) · Zbl 0926.65090
[19] Karni, S; Kurganov, A, Local error analysis for approximate solutions of hyperbolic conservation laws, Adv. Comput. Math., 22, 79-99, (2005) · Zbl 1127.65070
[20] Ketcheson, DI; Parsani, M; LeVeque, RJ, High-order wave propagation algorithms for hyperbolic systems, SIAM J. Sci. Comput., 35, a351-a377, (2013) · Zbl 1264.65151
[21] Kirby, R, On the convergence of high resolution methods with multiple time scales for hyperbolic conservation laws, Math. Comput., 72, 1239-1250, (2003) · Zbl 1018.65112
[22] Kolb, O, On the full and global accuracy of a compact third order WENO scheme, SIAM J. Numer. Anal., 52, 2335-2355, (2014) · Zbl 1408.65062
[23] Levy, D; Puppo, G; Russo, G, Compact central WENO schemes for multidimensional conservation laws, SIAM J. Sci. Comput., 22, 656-672, (2000) · Zbl 0967.65089
[24] Li, W; Ren, YX, High-order k-exact weno finite volume schemes for solving gas dynamic Euler equations on unstructured grids, Int. J. Numer. Methods Fluids, 70, 742-763, (2012)
[25] Lörcher, F; Gassner, G; Munz, CD, A discontinuous Galerkin scheme based on a space-time expansion. I. inviscid compressible flow in one space dimension, J. Sci. Comput., 32, 175-199, (2007) · Zbl 1143.76047
[26] Mandli, K.T., Ketcheson, D.I., et al.: Pyclaw software (2011). http://numerics.kaust.edu.sa/pyclaw · Zbl 0926.65090
[27] Ohlberger, M, A review of a posteriori error control and adaptivity for approximations of non-linear conservation laws, Int. J. Numer. Methods Fluids, 59, 333-354, (2009) · Zbl 1185.65163
[28] Osher, S; Sanders, R, Numerical approximations to nonlinear conservation laws with locally varying time and space grids, Math. Comput., 41, 321-336, (1983) · Zbl 0592.65068
[29] Puppo, G.: Numerical entropy production for central schemes. SIAM J. Sci. Comput. 25(4), 1382-1415 (2003/04) · Zbl 1061.65094
[30] Puppo, G; Semplice, M, Numerical entropy and adaptivity for finite volume schemes, Commun. Comput. Phys., 10, 1132-1160, (2011) · Zbl 1373.76140
[31] Puppo, G., Semplice, M.: Well-balanced high order 1d schemes on non-uniform grids and entropy residuals. Preprint arXiv:1403.4112 · Zbl 1371.65093
[32] Rogerson, A; Meiburg, E, A numerical study of the convergence properties of ENO schemes, J. Sci. Comput., 5, 151-167, (1990) · Zbl 0732.65086
[33] Semplice, M., Coco, A.: dune-fv software. http://www.personalweb.unito.it/matteo.semplice/codes.htm
[34] Shi, J; Hu, C; Shu, CW, A technique of treating negative weights in WENO schemes, J. Comput. Phys., 175, 108-127, (2002) · Zbl 0992.65094
[35] Shu, C.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (Cetraro, 1997), Lecture Notes in Mathematics, vol. 1697, pp. 325-432. Springer, Berlin (1998) · Zbl 0927.65111
[36] Shu, CW; Osher, S, Efficient implementation of essentially nonoscillatory shock-capturing schemes, J. Comput. Phys., 77, 439-471, (1988) · Zbl 0653.65072
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