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Palindromic discontinuous Galerkin method. (English) Zbl 1365.76251
Cancès, Clément (ed.) et al., Finite volumes for complex applications VIII – hyperbolic, elliptic and parabolic problems. FVCA 8, Lille, France, June 12–16, 2017. Cham: Springer (ISBN 978-3-319-57393-9/hbk; 978-3-319-57394-6/ebook; 978-3-319-58818-6/set). Springer Proceedings in Mathematics & Statistics 200, 171-178 (2017).
Summary: We present a high-order scheme for approximating kinetic equations with stiff relaxation. The construction is based on a high-order, implicit, upwind Discontinuous Galerkin formulation of the transport equations. In practice, because of the triangular structure of the implicit system, the computations are explicit. High order in time is achieved thanks to a palindromic composition method. The whole method is asymptotic-preserving with respect to the stiff relaxation and remains stable even with large CFL numbers.
For the entire collection see [Zbl 1371.65001].

MSC:
76M28 Particle methods and lattice-gas methods
65L04 Numerical methods for stiff equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
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References:
[1] Aregba-Driollet, D., Natalini, R.: Discrete kinetic schemes for multidimensional systems of conservation laws. SIAM J. Numer. Anal. 37(6), 1973–2004 (2000) · Zbl 0964.65096
[2] Badwaik, J., Boileau, M., Coulette, D., Franck, E., Helluy, P., Mendoza, L., Oberlin, H.: Task-based parallelization of an implicit kinetic scheme. arXiv preprint arXiv:1702.00169
(2017)
[3] Bey, J., Wittum, G.: Downwind numbering: robust multigrid for convection-diffusion problems. Appl. Numer. Math. 23(1), 177–192 (1997) · Zbl 0879.65088
[4] Bouchut, F.: Construction of BGK models with a family of kinetic entropies for a given system of conservation laws. J. Stat. Phys. 95(1–2), 113–170 (1999) · Zbl 0957.82028
[5] Chen, S., Doolen, G.D.: Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30(1), 329–364 (1998) · Zbl 1398.76180
[6] Coquel, F., Nguyen, Q.L., Postel, M., Tran, Q.H.: Large time step positivity-preserving method for multiphase flows. In: Hyperbolic Problems: Theory, Numerics, Applications, pp. 849–856. Springer, Heidelberg (2008) · Zbl 1138.65068
[7] Coulette, D., Franck, E., Helluy, P., Mehrenberger, M., Navoret, L.: Palindromic discontinuous Galerkin method for kinetic equations with stiff relaxation. arXiv preprint arXiv:1612.09422
(2016) · Zbl 1365.76251
[8] Graille, B.: Approximation of mono-dimensional hyperbolic systems: a lattice Boltzmann scheme as a relaxation method. J. Comput. Phy. 266, 74–88 (2014) · Zbl 1310.76145
[9] Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration: structure-preserving algorithms for ordinary differential equations, vol. 31. Springer Science & Business Media (2006) · Zbl 1094.65125
[10] Kahan, W., Li, R.C.: Composition constants for raising the orders of unconventional schemes for ordinary differential equations. Math. Comput. Am. Math. Soc. 66(219), 1089–1099 (1997) · Zbl 0870.65060
[11] McLachlan, R.I., Quispel, G.R.W.: Splitting methods. Acta Numer. 11, 341–434 (2002) · Zbl 1105.65341
[12] Shi, X., Lin, J., Yu, Z.: Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element. Int. J. Numer. Methods Fluids 42(11), 1249–1261 (2003) · Zbl 1033.76046
[13] Suzuki, M.: Fractal decomposition of exponential operators with applications to many-body theories and Monte Carlo simulations. Phys. Lett. A 146(6), 319–323 (1990)
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