Low regularity exponential-type integrators for the “good” Boussinesq equation

Abstract

Abstract In this paper, two semidiscrete low regularity exponential-type integrators are proposed and analyzed for the “good” Boussinesq equation, including a first-order integrator and a second-order one. Compared to the existing numerical methods, the convergence rate can be achieved under weaker regularity assumptions on the exact solution. Specifically, the first-order integrator is convergent linearly in $H^r$ for solutions in $H^{r+1}$ if $r>1/2$, i.e., the boundedness of one additional derivative of the solution is required to achieve the first-order convergence. When $r>7/6$, we can even prove linear convergence in $H^r$ for solutions in $H^{r+2/3}$. What’s more, half-order convergence is established in $H^{r}(r>3/2)$ for any solutions in $H^r$, i.e., no additional smoothness assumptions are needed. For the second-order integrator, the quadratic convergence in $H^{r}$$(r>1/2)$ (or $L^2$) is demonstrated, when the solutions belong to $H^{r+2}$ (or $H^{9/4}$). Numerical examples illustrating the convergence analysis are included. A comparison with other methods demonstrates the superiority of the newly proposed exponential-type integrators for rough data.