Error estimates for the AEDG method to one-dimensional linear convection-diffusion equations

Abstract:

We study the error estimates for the alternating evolution discontinuous Galerkin (AEDG) method to one-dimensional linear convection-diffusion equations. The AEDG method for general convection-diffusion equations was introduced by H. Liu and M. Pollack [J. Comp. Phys. 307 (2016), 574–592], where stability of the semi-discrete scheme was rigorously proved for linear problems under a CFL-like stability condition $\epsilon < Qh^2$. Here $\epsilon$ is the method parameter, and $h$ is the maximum spatial grid size. In this work, we establish optimal $L^2$ error estimates of order $O(h^{k+1})$ for $k$-th degree polynomials, under the same stability condition with $\epsilon \sim h^2$. For a fully discrete scheme with the forward Euler temporal discretization, we further obtain the $L^2$ error estimate of order $O(\tau +h^{k+1})$, under the stability condition $c_0\tau \le \epsilon < Qh^2$ for time step $\tau$; and an error of order $O(\tau ^2+h^{k+1})$ for the Crank-Nicolson time discretization with any time step $\tau$. Key tools include two approximation spaces to distinguish overlapping polynomials, two bi-linear operators, coupled global projections, and a duality argument adapted to the situation with overlapping polynomials.