A Locally Conservative Eulerian–Lagrangian Finite Difference Method for the forced KdV equation

Abstract We propose a Locally Conservative Eulerian–Lagrangian Finite Difference Method (fdLCELM) for approximating the solution of the forced Korteweg–de Vries equation. The new numerical method employs an operator splitting scheme that solves the nonlinear transport equation and the linear dispersive equation sequentially. In order to conserve mass in the transport fractional step, we trace back each grid cell in time along the integral curve. The dispersive fractional step will be solved using a cell-centered finite difference method. Numerical examples are provided to confirm and illustrate the accuracy and mass conservation property of the new method. We also study the numerical stability and time evolution of various stationary solitary wave solutions in the presence of one or two bumps.