Efficient and Accurate Numerical Methods for Long-Wave Short-Wave Interaction Equations in the Semiclassical Limit Regime (Journal of Computational Mathematics, 2019)
This paper focuses on performance of several efficient and accurate numerical methodsfor the long-wave short-wave interaction equations in the semiclassical limit regime. Thekey features of the proposed methods are based on: (i) the utilization of the first-order orsecond-order time-splitting method to the nonlinear wave interaction equations; (ii) the application of Fourier pseudo-spectral method or compact finite difference approximation tothe linear subproblem and the spatial derivatives; (iii) the adoption of the exact integrationof the nonlinear subproblems and the ordinary differential equations in the phase space.The numerical methods under study are efficient, unconditionally stable and higher-orderaccurate, they are proved to preserve two invariants including the position density in $L^1$.Numerical results are reported for case studies with different types of initial data, theseresults verify the conservation laws in the discrete sense, show the dependence of the numerical solution on the time-step, mesh-size and dispersion parameter ε, and demonstratethe behavior of nonlinear dispersive waves in the semi-classical limit regime.
Keywords: Long-wave short-wave interaction equations; Semiclassical limit; Time-splitting method; Spectral method; Compact finite difference method; Conservative properties.