Error estimates of the third order runge-kutta alternating evolution discontinuous galerkin method for convection-diffusion problems

Liu, Hailiang; Wen, Hairui

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1709-1732.

In this paper, we present the stability analysis and error estimates for the alternating evolution discontinuous Galerkin (AEDG) method with third order explicit Runge-Kutta temporal discretization for linear convection-diffusion equations. The scheme is shown stable under a CFL-like stability condition $c_0\tau \le ϵ\le c_1{h}^2$. Here $ϵ$ is the method parameter, and $h$ is the maximum spatial grid size. We further obtain the optimal $L^2$ error of order $O({\tau }^3+h^{k+1})$. Key tools include two approximation finite element spaces to distinguish overlapping polynomials, coupled global projections, and energy estimates of errors. For completeness, the stability analysis and error estimates for second order explicit Runge-Kutta temporal discretization is included in the appendix.

Classification : 65M15, 65M60, 35K20
Mots clés : Alternating evolution, convection-diffusion equation, discontinuous Galerkin, error estimates, Runge-Kutta method

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