Geodesic
Stability analysis and error estimates of Lax–Wendroff discontinuous Galerkin methods for linear conservation laws
Sun, Zheng1 ; Shu, Chi-Wang1 
1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 3, pp. 1063-1087

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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 τ≤λh (where τ and h are the time step and the maximum element length respectively and λ>0 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 L 2 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 u and its first order time derivative u t in one dimension, and numerical examples are given to validate our analysis.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016049
Classification : 65M12, 65M15, 65M60
Keywords: Discontinuous Galerkin method, Lax–Wendroff time discretization, linear conservation laws, L2-stability, error estimates

Sun, Zheng 1 ; Shu, Chi-Wang 1

1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA. 
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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

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