Numerical Methods for a 2D “Bad” Boussinesq Equation: RK4, Strang Splitting, and High-frequency Fourier Modes

arXiv:2511.20666v1 Announce Type: new
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 are based on Runge-Kutta fourth order (RK4) and 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 solution with good accuracy. Including even just a few number of Fourier modes that violate the condition result in a blow-up solution. The accuracy of the method is measured by computing the $L^{infty}$ error against a soliton exact solution. The errors resulting from RK4 and Strang splitting differ slightly, with the RK4 performs insignificantly better. Using our numerical methods, we also display a simulation with Dirichlet boundary condition to account for wave reflections.