保证平流扩散方程非负系数的三阶数值通量格式

K. Sakai, D. Watabe
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引用次数: 1

摘要

根据平流方程数值计算的Godunov定理,在精度超过一阶的多项式格式族中,不存在具有常正差分系数的高阶格式。对于平流扩散方程,迄今为止还没有在一组超过二阶精度的数值格式中发现具有正差分系数的稳定格式。为了保证平流扩散方程有限差分方程的差分系数不为负,我们提出了一种三阶数值通量的计算格式。对该方案进行了优化,使数值通量的截断误差最小,同时满足随局部科朗数和扩散数而变化的差分系数的正性条件。该优化方案的特点是与传统的三阶方案(如KAWAMURA方案和UTOPIA方案)使用相同的模板号,在任何地方都保持三阶精度,而不需要任何数值通量限制器。我们将这种方法推广到多维方程中。对线性和非线性平流扩散方程进行了数值实验,验证了该格式对非线性Burger方程的适用性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A Third-Order Scheme for Numerical Fluxes to Guarantee Non-Negative Coefficients for Advection-Diffusion Equations
According to Godunov theorem for numerical calculations of advection equations, there exist no high-er-order schemes with constant positive difference coefficients in a family of polynomial schemes with an accuracy exceeding the first-order. In case of advection-diffusion equations, so far there have been not found stable schemes with positive difference coefficients in a family of numerical schemes exceeding the second-order accuracy. We propose a third-order computational scheme for numerical fluxes to guarantee the non-negative difference coefficients of resulting finite difference equations for advection-diffusion equations. The present scheme is optimized so as to minimize truncation errors for the numerical fluxes while fulfilling the positivity condition of the difference coefficients which are variable depending on the local Courant number and diffusion number. The feature of the present optimized scheme consists in keeping the third-order accuracy anywhere without any numerical flux limiter by using the same stencil number as convemtional third-order shemes such as KAWAMURA and UTOPIA schemes. We extend the present method into multi-dimensional equations. Numerical experiments for linear and nonlinear advection-diffusion equations were performed and the present scheme’s applicability to nonlinear Burger’s equation was confirmed.
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