用物理黏性构建高阶不振荡对流扩散差分格式

利用数值摄动算法, 通过扩散格式数值摄动重构把对流扩散方程的2阶中心差分格式(2-CDS)重构为高精度高分辨率格式, 解析分析和模型方程计算证实了新格式的高精度不振荡性质. 新格式是把物理黏性使流动光滑化的扩散运动规律引入2-CDS 中的结果. 该法显然与构建高级离散格式的常见方法不同. 证实: 数值摄动重构中引入扩散运动规律的结果格式与引入对流运动规律(下游不影响上游的规律)的结果格式一致, 说明对离散方程的数值摄动运算, 在维持原格式结构形式不动的条件下, 不仅能提高格式精度和稳健性, 且可揭示对流离散运动规律与扩散离散运动规律之间的内在关联;同时证实, 文中提出和使用的上、下游分裂方法是构建高精度不振荡离散格式的一个有效方法.

HIGHER-ORDER ACCURATE, NON-OSCILLATORY, THREE-NODES CENTRAL DIFFERENCE SCHEME FOR THE CONVECTIVE-DIFFUSION EQUATION CONSTRUCTED BY USING PHYSICAL VISCOSITY

Several higher-order accurate,non-oscillatory,three-nodes central difference schemes for the convective-diffusion equation are given by perturbationally reconstructing the diffusion scheme in the second-order accurate central difference scheme（2-CDS）.Excellent properties of higher-order accurate and high resolution of the present new scheme’s（diffusion perturbation schemes,DPS） are verified by theoertical analyses and three numerical tests which include one-dimensional linear and non-linear and two-dimensional convective-diffusion equations.In all numerical tests,the 2-CDS oscillates and diverges on coarse grids,while part of DPS do not oscillates and can capture discontinuities with high resolution.The mean square root L₂ errors of all DPS are greatly less than those of 2-CDS in all numerical tests.The DPS are the results of introducing diffusion-motion law（i.e.physical viscosity smoothing out space-distribution of diffusion quantities） into 2-CDS. The present method is obviously different from the well-known those of constructing high-order accurate and high resolution schemes.In addition,we prove that DPS are completely consistent with those schemes of introducing convection-motion law（i.e.law of that the downstream does not affect the upstream） into 2-CDS,to show that the perturbational operation to 2-CDS not only raises the scheme’s accurate and stability but also reveals intrinsic relation between the convective discrete scheme and diffusion discrete scheme,and that the upstream-downstream splitting is a very useful method for reconstructing high-order accurate,high resolution CFD scheme without artificial viscosity or limiter.