In this paper, we analyze the Lax–Wendroff discontinuous Galerkin (LWDG) method for solving linear conservation laws. The method was originally proposed by Guo et al. in [W. Guo, J.-M. Qiu and J. Qiu, J. Sci. Comput. 65 (2015) 299–326], where they applied local discontinuous Galerkin (LDG) techniques to approximate high order spatial derivatives in the Lax–Wendroff time discretization. We show that, under the standard CFL condition (where and are the time step and the maximum element length respectively and is a constant) and uniform or non-increasing time steps, the second order schemes with piecewise linear elements and the third order schemes with arbitrary piecewise polynomial elements are stable in the norm. The specific type of stability may differ with different choices of numerical fluxes. Our stability analysis includes multidimensional problems with divergence-free coefficients. Besides solving the equation itself, the LWDG method also gives approximations to its time derivative simultaneously. We obtain optimal error estimates for both the solution and its first order time derivative in one dimension, and numerical examples are given to validate our analysis.
Accepté le :
DOI : 10.1051/m2an/2016049
Mots clés : Discontinuous Galerkin method, Lax–Wendroff time discretization, linear conservation laws, L2-stability, error estimates
@article{M2AN_2017__51_3_1063_0, author = {Sun, Zheng and Shu, Chi-Wang}, title = {Stability analysis and error estimates of {Lax{\textendash}Wendroff} discontinuous {Galerkin} methods for linear conservation laws}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {1063--1087}, publisher = {EDP-Sciences}, volume = {51}, number = {3}, year = {2017}, doi = {10.1051/m2an/2016049}, zbl = {1373.65063}, mrnumber = {3666657}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2016049/} }
TY - JOUR AU - Sun, Zheng AU - Shu, Chi-Wang TI - Stability analysis and error estimates of Lax–Wendroff discontinuous Galerkin methods for linear conservation laws JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 1063 EP - 1087 VL - 51 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016049/ DO - 10.1051/m2an/2016049 LA - en ID - M2AN_2017__51_3_1063_0 ER -
%0 Journal Article %A Sun, Zheng %A Shu, Chi-Wang %T Stability analysis and error estimates of Lax–Wendroff discontinuous Galerkin methods for linear conservation laws %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 1063-1087 %V 51 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016049/ %R 10.1051/m2an/2016049 %G en %F M2AN_2017__51_3_1063_0
Sun, Zheng; Shu, Chi-Wang. Stability analysis and error estimates of Lax–Wendroff discontinuous Galerkin methods for linear conservation laws. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 1063-1087. doi : 10.1051/m2an/2016049. http://www.numdam.org/articles/10.1051/m2an/2016049/
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