Optimal error bounds of the time-splitting sine-pseudospectral method for the biharmonic nonlinear Schrödinger equation

Abstract

We propose a time-splitting sine-pseudospectral (TSSP) method for the biharmonic nonlinear Schrödinger equation (BNLS) and establish its optimal error bounds. In the proposed TSSP method, we adopt the sine-pseudospectral method for spatial discretization and the second-order Strang splitting for temporal discretization. The proposed TSSP method is explicit and structure-preserving, such as time symmetric, mass conservation and maintaining the dispersion relation of the original BNLS in the discretized level. Under the assumption that the solution of the one dimensional BNLS belongs to $H^m$ with $m\ge 9$, we prove error bounds at $O({\tau }^2+h^m)$ and $O({\tau }^2+h^{m-1})$ in $L^2$ norm and $H^1$ norm respectively, for the proposed TSSP method, with τ time step and h mesh size. For general dimensional cases with $d=1,2,3$, the error bounds are at $O({\tau }^2+h^m)$ and $O({\tau }^2+h^{m-2})$ in $L^2$ and $H^2$ norm under the assumption that the exact solution is in $H^m$ with $m\ge 10$. The proof is based on the bound of the Lie-commutator for the local truncation error, discrete Gronwall inequality, energy method and the $H^1$- or $H^2$-bound of the numerical solution which implies the $L^{\infty }$-bound of the numerical solution. Finally, extensive numerical results are reported to confirm our optimal error bounds and to demonstrate rich phenomena of the solutions including rapidly dispersion in space of high frequency waves and soliton collisions.