Error self-canceling of a difference scheme maintaining two conservation laws for linear advection equation
Error self-canceling of a difference scheme maintaining two conservation laws for linear advection equation
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DOI：
10.1090/s0025-5718-2011-02523-8
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发表时间：
2011-07
期刊：
影响因子：
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通讯作者：
Yanfen Cui;De-kang Mao
中科院分区：
文献类型：
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作者：
Yanfen Cui;De-kang Mao
。近年来，毛和他的同事开发了一种新型的演化偏微分方程差分格式。新方案的核心是除了方程的原始未知数之外，还模拟一些作为未知数的非线性函数的量；因此，它们保留了方程的附加非线性离散结构。该方案表现出超收敛性，其数值解在精度和长时间行为方面都远远优于传统差分方案。在本文中，为了理解该格式的超收敛性质，我们对维持线性平流方程两个守恒定律的格式进行了截断误差研究。该方案是此类方案中最简单的一种。我们的研究表明，该方案在不同时间步长中产生的数值误差以非线性方式累积，并且相互抵消。据我们所知，这种误差自抵消特性在其他数值方法中还没有见过，正是这个特性带来了该方案的超收敛性。
. 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.