We present an algorithm to solve the dispersive depth-averaged Serre-Green-Naghdi (SGN) equations using patch-based adaptive mesh refinement. These equations require adding additional higher derivative terms to the nonlinear shallow water equations. This has been implemented as a new component of the open source GeoClaw software that is widely used for modeling tsunamis, storm surge, and related hazards, improving its accuracy on shorter wavelength phenomena. The equations require the solution of an elliptic system at each time step. The adaptive algorithm allows different time steps on different refinement levels, and solves the implicit equations level by level. Computational examples are presented to illustrate the stability and accuracy on a radially symmetric test case and two realistic tsunami modeling problems, including a hypothetical asteroid impact creating a short wavelength tsunami for which dispersive terms are necessary.

翻译：我们提出了一种基于块自适应网格细化的算法，用于求解色散深度平均Serre-Green-Naghdi (SGN)方程。这些方程需要在非线性浅水方程中添加额外的高阶导数项。该算法已作为开源软件GeoClaw的新组件实现——该软件广泛应用于海啸、风暴潮及相关灾害的建模，从而提升了其在短波长现象上的计算精度。该方程需要在每个时间步求解一个椭圆系统。自适应算法允许不同细化层级采用不同时间步长，并逐层级求解隐式方程。我们通过计算实例验证了该算法的稳定性与精度：包括一个径向对称测试案例，以及两个实际海啸建模问题——其中一个涉及产生短波长海啸的假定小行星撞击事件，此时色散项是不可或缺的。