求解二维浅水方程的高分辨率完全松弛格式(英文)

High-resolution Relaxed Scheme for Two-dimensional Shallow Water Equations

浅水方程在水利、海洋和环境工程等领域都具有重要的应用.对二维浅水方程,提出了一种高分辨率的完全松弛格式.该格式以松弛近似方法和改进的五阶WENO重构为基础,重构方法的引入提高了格式的精度,并保证格式的基本无振荡性质.时间方向的离散采用三阶保持强稳定性质的Runge-Kutta方法.该格式保持了完全松弛格式简单的优点,即避免了利用Riemann解算器和计算通量函数的雅可比矩阵.用数值算例对格式进行了检验,结果表明该方法健全、有效.

The shallow water equations have important applications in hydraulic, ocean and envi- ronmental engineering. A high-resolution relaxed scheme for approximating solutions of two-dimensional shallow water equations is presented. The scheme is based on the re- laxation approximation and an improved fifth-order weighted essentially non-osscillatory （WENO） reconstruction. This reconstruction is adopted in order to improve the accu- racy and to guarantee the non-oscillatory behavior of the resulting method. A third-order strong stability preserving （SSP） Runge-Kutta scheme is used for the time discretization. The scheme benefits from the simplicity of relaxed schemes in that it avoids Riemann solvers and the computation of Jacobians. The performance of our method is illustrated by several numerical experiments. The results show that it is efficient and robust.