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. We use a formulation that requires solving an elliptic system of equations at each time step, making the method implicit. 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：我们提出一种基于分块自适应网格细化的算法，用于求解色散深度平均的Serre-Green-Naghdi (SGN)方程。这类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GeoClaw软件的新组件实现——该软件广泛用于海啸、风暴潮及相关灾害模拟，通过本算法可提升其对短波长现象的模拟精度。我们采用一种需要在每个时间步求解椭圆方程组的公式化方法，使算法具有隐式特性。自适应算法允许在不同细化层级使用不同时间步长，并逐层求解隐式方程。通过径向对称测试案例及两个真实海啸建模问题（包括需引入色散项的假想小行星撞击引发的短波长海啸）的计算示例，验证了算法的稳定性与精度。