Commun. Comput. Phys., 6 (2009), pp. 203-230.


A Family of Characteristic Discontinuous Galerkin Methods for Transient Advection-Diffusion Equations and Their Optimal-Order L^2 Error Estimates

Kaixin Wang 1, Hong Wang 2*, Mohamed Al-Lawatia 3, Hongxing Rui 1

1 School of Mathematics and System Sciences, Shandong University, Jinan, Shandong 250100, China.
2 Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208, USA.
3 Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Al-Khod 123, Sultanate of Oman.

Received 20 November 2007; Accepted (in revised version) 25 October 2008
Available online 24 November 2008

Abstract

We develop a family of characteristic discontinuous Galerkin methods for transient advection-diffusion equations, including the characteristic NIPG, OBB, IIPG, and SIPG schemes. The derived schemes possess combined advantages of Eulerian-Lagrangian methods and discontinuous Galerkin methods. An optimal-order error estimate in the L^2 norm and a superconvergence estimate in a weighted energy norm are proved for the characteristic NIPG, IIPG, and SIPG scheme. Numerical experiments are presented to confirm the optimal-order spatial and temporal convergence rates of these schemes as proved in the theorems and to show that these schemes compare favorably to the standard NIPG, OBB, IIPG, and SIPG schemes in the context of advection-diffusion equations.

AMS subject classifications: 35R32, 37N30, 65M12, 76N15

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Key words: Advection-diffusion equation, characteristic method, discontinuous Galerkin method, numerical analysis, optimal-order L^2 error estimate, superconvergence estimate.

*Corresponding author.
Email: kx.wang@mail.sdu.edu.cn (K. Wang), hwang@math.sc.edu (H. Wang), allawati@squ.edu.om (M. Al-Lawatia), hxrui@sdu.edu.cn (H. Rui)
 

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