Numerical Methods for a Two-Dimensional "Bad" Boussinesq Equation by Removal of High-frequency Fourier Modes

Abstract

Numerical methods for a two-dimensional "bad" Boussinesq equation: $u_{tt} = u_{xx} + u_{xxxx} + u_{yy} - 3 (u^{2})_{xx}$ are presented with good accuracy. The methods mainly depend on pseudo-spectral Fourier with a trimming of carefully chosen high-frequency Fourier modes. One method also relies on Runge-Kutta fourth order (RK4), and another method relies on Strang operator splitting. Before implementing the two methods, we analyze using Fourier series the linearized version of the equation by removing the nonlinear term $3(u^{2})_{xx}$, and found that a particular bound or condition needs to be satisfied to avoid blow-up solution. We found that high-frequency Fourier modes that do not satisfy the condition must be excluded from the Fourier solution. We then apply this condition to the numerical methods for solving the nonlinear Boussinesq equation and found that including only the Fourier modes that satisfy the condition gives stable numerical solution with good accuracy up to $t=100$. Including even just a few number of Fourier modes that violate the condition can result in a blow-up solution as early as $t=23.5$. The accuracy of the method is measured by computing the $L^{\infty}$ error against a soliton exact solution. The errors resulting from Runge-Kutta fourth order and Strang splitting numerical simulations differ slightly for small $\triangle t$, while there is a noticeable decrease in performance for the Strang splitting simulation as $\triangle t$ increases. Using our numerical methods, we also display a simulation with Dirichlet boundary condition to account for wave reflections.