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

TL;DR

This work addresses numerical solution of the two-dimensional \\textit{bad} Boussinesq equation by deriving a linear stability trimming condition for Fourier modes and implementing two spectral schemes: RK4 with a pseudo-spectral discretization and Strang operator splitting. Both methods rely on a mode-trimming mask to exclude unstable high-frequency components, enabling stable, accurate simulations of the nonlinear equation against a soliton reference. RK4 shows a slight accuracy advantage over Strang splitting, and enforcing the trimming condition is essential to prevent blow-up, even when only a few high-frequency modes are violated. The study also demonstrates Dirichlet boundary reflections and provides practical, stable tools for simulating ill-posed dispersive PDEs in 2D.

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.

Figures (9)

• Figure 1: Comparison of the function $F(j,k) =L - \frac{\pi j^{2}}{\sqrt{k^{2}+j^{2}}}$ when $L=100$ and $L=200$. The red lines are where $F(j,k)=0$.
• Figure 2: The colormap plot of the numerical solution at $t=40$ for the first simulation via Runge-Kutta method (left). The red dashed lines indicate $y=-5$ and $y=5$, we only compute the error in between these lines.
• Figure 3: The colormap plot of the numerical solution at $t=40$ for the first simulation via Strang splitting method (right). The red dashed lines indicate $y=-5$ and $y=5$, we only compute the error in between these lines.
• Figure 4: The $L^{\infty}$ error for both numerical methods up to $t=40$ for the first simulation.
• Figure 5: Surface plots of the blow-up numerical solutions at $t=23.5$ when we minorly violate the stable condition (\ref{['stable_condition_2']}).