TVD格式求解对流扩散方程的最优限制器研究

Examination of Optimal Flux Limiters in TVD Scheme for Solving Convection-Diffusion Equation

为探究不同通量限制器应用于TVD(Total Variation Diminishing)格式求解对流扩散方程时的适用性,基于3种典型的TVD格式与10种常用的通量限制器,分别求解了线性对流扩散方程、非线性对流扩散方程、拟线性对流扩散方程。数值结果表明,相比于MUSCL(Monotonic Upstream-centered Scheme for Conservation Laws)和MTVDLF(Modified TVDLF)格式,采用TVDLF(TVD Lax-Friedrichs)格式时,计算结果出现了较为严重的数值耗散;对MUSCL和MTVDLF格式进行具体分析发现,关于阶跃型纯对流问题,Superbee限制器的误差最小,Minmod误差最大。关于高斯型对流扩散问题,Minmod误差最大,Woodward误差最小。而关于阶跃型对流扩散问题及Burgers方程,限制器的类型对实验结果影响并不明显。

In order to explore the applicability of different flux limiters applied to solve convection-diffusion equations in TVD(Total Variation Diminishing)scheme,based on 3 typical TVD schemes and 10 commonly used flux limiters,this study solved the linear convection-diffusion equation,the nonlinear convection-diffusion equation and the quasi-linear convection-diffusion equation respectively.The results show that:compared with MUSCL(Monotonic Upstream-Centered Scheme for Conservation Laws)and MTVDLF(Modified TVDLF),when the TVDLF(TVD Lax-Friedrichs)scheme is used,the numerical dissipation of the calculation results is quite serious.By analyzing MUSCL and MTVDLF schemes,it is found that the error of the Superbee limiter is the smallest and the error of Minmod is the largest for the step type pure convection problem.For the Gaussian convection-diffusion problem,the error of Minmod is the largest and Woodward is the smallest.For the step convection-diffusion problem and Burgers’equation,the type of limiter has no obvious effect on the solutions.