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Explicit, unconditionally stable, high-order schemes for the approximation of some first- and second-order linear, time-dependent partial differential equations (PDEs) are proposed. The schemes are based on a weak formulation of a semi-Lagrangian scheme using discontinuous Galerkin (DG) elements. It follows the ideas of the recent works of Crouseilles et al. [N. Crouseilles, M. Mehrenberger and F. Vecil, In CEMRACS’10 research achievements: numerical modeling of fusion. ESAIM Proc. 32 (2011) 211–230], Rossmanith and Seal [J.A. Rossmanith and D.C. Seal, J. Comput. Phys. 230 (2011) 6203–6232], for first-order equations, based on exact integration, quadrature rules, and splitting techniques for the treatment of two-dimensional PDEs. For second-order PDEs the idea of the scheme is a blending between weak Taylor approximations and projection on a DG basis. New and sharp error estimates are obtained for the fully discrete schemes and for variable coefficients. In particular we obtain high-order schemes, unconditionally stable and convergent, in the case of linear first-order PDEs, or linear second-order PDEs with constant coefficients. In the case of non-constant coefficients, we construct, in some particular cases, “almost” unconditionally stable second-order schemes and give precise convergence results. The schemes are tested on several academic examples.
Bokanowski, Olivier 1, 2 ; Simarmata, Giorevinus 3
@article{M2AN_2016__50_6_1699_0,
author = {Bokanowski, Olivier and Simarmata, Giorevinus},
title = {Semi-Lagrangian discontinuous {Galerkin} schemes for some first- and second-order partial differential equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1699--1730},
publisher = {EDP-Sciences},
volume = {50},
number = {6},
year = {2016},
doi = {10.1051/m2an/2016004},
zbl = {1357.65171},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2016004/}
}
TY - JOUR AU - Bokanowski, Olivier AU - Simarmata, Giorevinus TI - Semi-Lagrangian discontinuous Galerkin schemes for some first- and second-order partial differential equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 1699 EP - 1730 VL - 50 IS - 6 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2016004/ DO - 10.1051/m2an/2016004 LA - en ID - M2AN_2016__50_6_1699_0 ER -
%0 Journal Article %A Bokanowski, Olivier %A Simarmata, Giorevinus %T Semi-Lagrangian discontinuous Galerkin schemes for some first- and second-order partial differential equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 1699-1730 %V 50 %N 6 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2016004/ %R 10.1051/m2an/2016004 %G en %F M2AN_2016__50_6_1699_0
Bokanowski, Olivier; Simarmata, Giorevinus. Semi-Lagrangian discontinuous Galerkin schemes for some first- and second-order partial differential equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 6, pp. 1699-1730. doi: 10.1051/m2an/2016004
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