稳态双曲守恒律问题的快速扫描WCNS方法

Fast Sweeping WCNS for Steady State Problems for Hyperbolic Conservation Laws

非线性稳态双曲守恒律问题广泛存在于流体力学、优化控制、图像处理等领域,因此设计了求解此类问题的快速扫描加权紧致非线性格式(WCNS)。首先,在模型方程中引入时间导数项,将稳态问题转化为时间相关问题。方程通量导数采用三阶WCNS格式计算,时间导数采用三阶TVD Runge-Kutta方法计算。其次,结合Gauss-Seidel型迭代方法和有限次交替扫描顺序的快速扫描策略用于加快算法收敛速度。最后数值结果表明:快速扫描WCNS方法能达到三阶设计精度且具有很好的激波捕捉能力;相比未引入快速扫描策略的TVD Runge-Kutta WCNS方法,快速扫描WCNS方法收敛速度更快,计算效率更高。

Steady state problems for nonlinear hyperbolic conservation laws widely exist in many fields,such as fluid mechanics,optimal control,picture processing,and so on.A fast sweeping weighted compact nonlinear scheme(WCNS)is designed for these problems.The model equations are first transformed into the time-dependent problems by adding time derivative terms.Then,the flux derivatives are computed by a third-order WCNS and a third-order TVD Runge-Kutta method is used for the time discretization.Moreover,the fast sweeping strategy that combines Gauss-Seidel-type iteration method with a finite number of alternating sweeping orderings is adopted to accelerate the convergence speed of the designed algorithm.Numerical results show that the fast sweeping WCNS has third-order accuracy and good shockcapturing ability.Compared with the TVD Runge-Kutta WCNS without sweeping,the fast sweeping WCNS has a faster convergence speed and higher computational efficiency.