A Semi-Implicit Runge–Kutta Time-Difference Scheme for the Two-Dimensional Shallow-Water Equations

Abstract

A semi-implicit, two time-level, three-step iterative time-difference scheme is proposed for the two-dimensional nonlinear shallow-water equations in a conservative flux form. After a semi-implicit linearization of the governing equations, the linear gravity wave terms are time discretized implicitly using a second-order trapezoidal scheme applied over each iterative step, whereas the nonlinear terms including horizontal advection and other terms left over from the semi-implicit linearization are time discretized explicitly using a third-order Runge?Kutta scheme. The effectiveness of the scheme in terms of numerical accuracy, stability, and efficiency is established through a forced initial-boundary value problem studied using a two-dimensional shallow-water model.