粘性Burgers方程的高阶精度半隐式WCNS方法

HIGH-ORDER SEMI-IMPLICIT WCNS METHOD FOR BURGERS'EQUATION

Burgers方程为Navier-Stokes方程组的简化形式,在计算数学和计算流体力学领域中有着广泛应用.本文设计了粘性Burgers方程的高阶精度半隐式加权紧致非线性格式(WCNS),并给出了稳定性分析.方程对流项和粘性项分别采用五阶精度WCNS格式和四阶中心差分格式计算.半离散系统采用三阶精度IMEX Runge-Kutta方法计算,对流项和粘性项分别进行显式和隐式处理.数值结果表明IMEX Runge-Kutta WCNS格式可达到三阶时间精度和五阶空间精度,比显式TVD Runge-Kutta WCNS格式计算效率高,且具有高分辨率的激波捕捉能力.

Burgers'equations are a simplified form of incompressible Navier-Stokes equations and have been widely used in computational mathematics and computational fluid dynamics.This paper designs a high-order semi-implicit weighted compact nonlinear scheme(WCNS)for viscous Burgers'equations and gives the stability analysis of the designed scheme.The fifth-order WCNS and the fourth-order central difference scheme are used for the spatial discretization of convective terms and viscous terms.The third-order IMEX Runge-Kutta scheme is used for the time discretization of the semi-discrete system and convective terms are treated explicitly,while viscosity terms are treated implicitly.Numerical results show that the IMEX Runge-Kutta WCNS can achieve third-order accuracy in time and fifth-order accuracy in space.This semi-implicit WCNS is better than the TVD Runge-Kutta WCNS in terms of computational efficiency and has high-resolution shock-capturing ability.