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 | References | Similar Articles | Additional Information

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

DOI:
https://doi.org/10.1090/S0025-5718-2011-02523-8

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.