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

Authors: Hailiang Liu and Hairui Wen
Journal: Math. Comp. 87 (2018), 123-148
MSC (2010): Primary 65M15, 65M60, 35K20
Published electronically: May 1, 2017
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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.

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Additional Information

Hailiang Liu
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011

Hairui Wen
Affiliation: School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100080, People’s Republic of China

Keywords: Alternating evolution, convection-diffusion equations, discontinuous Galerkin, error estimates
Received by editor(s): April 5, 2016
Received by editor(s) in revised form: August 30, 2016
Published electronically: May 1, 2017
Article copyright: © Copyright 2017 American Mathematical Society