Analysis and application of a compact multistep ADI solver for a class of nonlinear viscous wave equations

In this article, a compact multistep alternating direction implicit (ADI) method is derived for solving a class of two-dimensional (2D) nonlinear viscous wave equations. This ADI method uses the combination of second-order backward differentiation formula (BDF2) solver with approximation factorization for time integration, and fourth-order Padé approximations to the second spatial derivatives for spatial discretization. It is shown by the discrete energy method that the present ADI method has good stability, and can attain convergence rate of O(Δt2+hx4+hy4) in L2- and H1-norms. Besides, the application of a three-grid Richardson extrapolation algorithm to the ADI solution can make final solution fourth-order accurate in both time and space. Numerical results are given to demonstrate the usefulness and efficiency of the resulting algorithms.