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Error self-canceling of a difference scheme maintaining two conservation laws for linear advection equation

Authors: Cui Yanfen and Mao De-kang
Journal: Math. Comp. 81 (2012), 715-741
MSC (2010): Primary 65M06, 65M15
Published electronically: July 26, 2011
MathSciNet review: 2869034
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Abstract: In recent years, Mao and his co-workers developed a new type of difference schemes for evolution partial differential equations. The core of the new schemes is to simulate, in addition to the original unknowns of the equations, some quantities that are nonlinear functions of the unknowns; therefore, they maintain additional nonlinear discrete structures of the equations. The schemes show a super-convergence property, and their numerical solutions are far better than that of traditional difference schemes at both accuracy and long-time behavior.

In this paper, to understand the super-convergence properties of the schemes, we carry out a truncation error investigation on the scheme maintaining two conservation laws for the linear advection equation. This scheme is the simplest one of this type. Our investigation reveals that the numerical errors of the scheme produced in different time steps are accumulated in a nonlinear fashion, in which they cancel each other. As to our knowledge, such an error self-canceling feature has not been seen in other numerical methods, and it is this feature that brings the super-convergence property of the scheme.

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Additional Information

Cui Yanfen
Affiliation: Shanghai Institute of Applied Mathematics and Mechanics, Shanghai, 200072, People’s Republic of China
Address at time of publication: Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

Mao De-kang
Affiliation: Department of Mathematics, Shanghai University, Shanghai, 200444, People’s Republic of China

Keywords: Scheme maintaining two conservation laws, truncation error, error self-canceling
Received by editor(s): March 14, 2009
Received by editor(s) in revised form: January 22, 2011
Published electronically: July 26, 2011
Additional Notes: This research was supported by China National Science Foundation Grant No.10971132 and Shanghai Pu Jiang Program [2006] 118, and also by Shanghai Leading Academic Discipline project (J50101)
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.