Two exponential-type integrators for the “good” Boussinesq equation. (English) Zbl 1428.35425
Summary: We introduce two exponential-type integrators for the “good” Boussinesq equation. They are of orders one and two, respectively, and they require lower spatial regularity of the solution compared to classical exponential integrators. For the first integrator, we prove first-order convergence in \(H^r\) for solutions in \(H^{r+1}\) with \(r>1/2\). This new integrator even converges (with lower order) in \(H^r\) for solutions in \(H^r\), i.e., without any additional smoothness assumptions. For the second integrator, we prove second-order convergence in \(H^r\) for solutions in \(H^{r+3}\) with \(r>1/2\) and convergence in \(L^2\) for solutions in \(H^3\). Numerical results are reported to illustrate the established error estimates. The experiments clearly demonstrate that the new exponential-type integrators are favorable over classical exponential integrators for initial data with low regularity.
MSC:
| 35Q40 | PDEs in connection with quantum mechanics |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 81Q05 | Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
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