A family of new fourth-order solvers for a nonlinear damped wave equation

In this paper, we propose a family of new three-level compact alternating direction implicit (ADI) difference schemes for solving a linear wave equation with a nonlinear damping function. By using a fourth-order accurate scheme to approximate the exact solution at the first time level, it is shown through the energy method that these difference schemes have fourth-order accuracy in space and second-order accuracy in time with respect to H1H1- and L∞L∞-norms. A class of Richardson extrapolation algorithms based on three time-grid parameters are presented to obtain approximate solution of fourth-order accuracy in both time and space in L∞L∞-norm. Numerical experiments are performed to support our theoretical results and test the accuracy and efficiency of our algorithms.