Modeling and Visualization of a Time-Dependent Shallow Water System Using Nonlinear Kirchhoff Circuit

A physics oriented numerical algorithm based on a nonlinear Kirchhoff circuit is detailed for the modeling and visualization of a two-dimensional time-dependent nonlinear shallow water (NLSW) system described by a set of partial differential equations (PDEs). In essence, the continuous physical system served by a nonlinear Kirchhoff circuit is transformed to an equivalent discrete dynamic system implemented by a multidimensional wave digital filtering (MDWDF) network. This amounts to numerically approximating the differential equations used to describe elements of a MD passive electrical circuit by grid-based difference equations. Details to achieve the desired digital network with initial and boundary conditions incorporated are described. In particular, the MD passivity established in the reference circuit has led to uniquely a high degree of robustness of the MDWDF architecture with the focus on the handling of computational errors. Visual demonstration is depicted for some nature effects of wave propagation from and across artificial boundaries in unbounded and bounded domains. An insight about numerical stability and error propagations directly linked to the passivity of the reference circuit is presented by extensively analyzing the proposed solution for the convergent behaviors of the MDWDF network. In addition, feasible comparisons between the MDWDF method and the finite element method (FEM) implemented in the COMSOL Multiphysics are also presented by focusing on the stability performance in a bounded domain.