Numerical Methods For The “Good” Boussinesq Equation

We consider different numerical methods for the “Good” Boussinesq(GB) equation. Firstly, we propose a numerical scheme with second order temporal accuracy by applying the operator splitting technique in time and pseudospectral formulation in space. We present the detailed theoretical analysis and some numerical experiments to validate the analytical results. As far as we know, this is the first time the operator splitting technique is applied to the GB equation. We further present two high temporal order algorithms for solving the GB equation. The main idea is to iteratively apply the low order methods to solve an error’s equation and refine the provisional solution until it converges to the high order pseudospectral solution both in space and time. This idea is further coupled with existing Jacobi-Free Newton Krylov(JFNK)method for improved efficiency. Unfortunately, a straigtforward formulation in the first algorithm has a numerical condition number of O(N4), where N is the number of Fourier terms used in the spatial direction. To further improve the numerical stability and accuracy, we present a second formulation where an integral equation approach is applied to the linear terms of the GB equation analytically. Our numerical experiments show that the second algorithm has condition number O(1) and can achieve machine precision accuracy.