We present an advection-pressure flux-vector splitting method for the one and two- dimensional shallow water equations following the approach first proposed by Toro and V\'azquez for the compressible Euler equations. The resulting first-order schemes turn out to be exceedingly simple, with accuracy and robustness comparable to that of the sophisticated Godunov upwind method used in conjunction with complete non- linear Riemann solvers. The technique splits the full system into two subsystems, namely an advection system and a pressure system. The sought numerical flux results from fluxes for each of the subsystems. The basic methodology, extended on 2D unstructured meshes, constitutes the building block for the construction of numerical schemes of very high order of accuracy following the ADER approach. The presented numerical schemes are systematically assessed on a carefully selected suite of test problems with reference solutions, in one and two space dimensions.The applicabil- ity of the schemes is illustrated through simulations of tsunami wave propagation in the Pacific Ocean.
翻译:我们按照Toro和V\'azquez首先为压缩 Euler 方程式提出的方法,为1和2维浅水方程式提出了一个对冲压通量分解方法。由此得出的第一级方案非常简单,其准确性和稳健性可与先进的Godunov上风方法相仿,该方法与完整的非线性里伊曼理算器一起使用。该技术将整个系统分为两个子系统,即吸附系统和压力系统。寻求每个子系统的通量所产生的数字通量结果。在2D非结构化的meshes上扩展的基本方法构成了在ADER 方法之后构建高度精准的数字方案的基础。所提出的数字方案是在经过仔细选择的一组测试问题中系统地评估的,其中包括一个和两个空间层面的参考解决方案。通过太平洋海啸波波传播模拟来说明这些办法的相近性。