标题： CFL优化的浅水方程前向后向龙格-库塔格式 显示英文标题

标题： CFL Optimized Forward-Backward Runge-Kutta Schemes for the Shallow Water Equations

摘要： 我们提出了一个用于求解浅水方程的龙格-库塔型时间推进方案的公式和优化，旨在显著增加相对于可比方法的有效允许时间步长。 该方案称为FB-RK(3,2)，利用厚度数据的加权前后向平均来推进动量方程。 这些权重通过采用冯·诺依曼型分析的优化过程进行选择，确保权重最大化允许的柯朗数。 通过简化的局部截断误差分析和数值实验，我们表明该方法对于任何权重选择至少是二阶的时间精度，并且对于充分解析的波具有低色散和耗散误差。 此外，我们表明优化后的FB-RK(3,2)在准线性测试案例中可以取到比一种流行的三阶段、三阶强稳定性保持龙格-库塔方法大2.8倍的时间步长。 在与海洋和大气流动相关的完全非线性浅水测试案例中，FB-RK(3,2)在允许的时间步长上比SSPRK3高出大约1.6到2.2倍，使该方案的计算效率大约提高一倍，而对解的质量几乎没有影响。 显示英文摘要

摘要： We present the formulation and optimization of a Runge-Kutta-type time-stepping scheme for solving the shallow water equations, aimed at substantially increasing the effective allowable time-step over that of comparable methods. This scheme, called FB-RK(3,2), uses weighted forward-backward averaging of thickness data to advance the momentum equation. The weights for this averaging are chosen with an optimization process that employs a von Neumann-type analysis, ensuring that the weights maximize the admittable Courant number. Through a simplified local truncation error analysis and numerical experiments, we show that the method is at least second order in time for any choice of weights and exhibits low dispersion and dissipation errors for well-resolved waves. Further, we show that an optimized FB-RK(3,2) can take time-steps up to 2.8 times as large as a popular three-stage, third-order strong stability preserving Runge-Kutta method in a quasi-linear test case. In fully nonlinear shallow water test cases relevant to oceanic and atmospheric flows, FB-RK(3,2) outperforms SSPRK3 in admittable time-step by factors between roughly between 1.6 and 2.2, making the scheme approximately twice as computationally efficient with little to no effect on solution quality.