Uniformly and Optimally Accurate Methods for the Zakharov System in the Subsonic Limit Regime

We present two uniformly and optimally accurate numerical methods for discretizing the Zakharov system (ZS) with a dimensionless parameter 0 < ε ≤ 1, which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e. 0 < ε ≪ 1, the solution of ZS propagates waves with O(ε)and O(1)-wavelength in time and space, respectively, and/or rapid outgoing initial layers with speed O(1/ε) in space due to the singular perturbation of the wave operator in ZS and/or the incompatibility of the initial data. By adopting an asymptotic consistent formulation of ZS, we present a time-splitting exponential wave integrator (TS-EWI) method via applying a time-splitting technique and an exponential wave integrator for temporal derivatives in the nonlinear Schrödinger equation and wave-type equation, respectively. By introducing a multiscale decomposition of ZS, we propose a time-splitting multiscale time integrator (TS-MTI) method. Both methods are explicit and they are uniformly and optimally accurate in space for all kind of initial data and ε ∈ (0, 1]. The TS-EWI method is simpler to be implemented and it is only uniformly and optimally accurate in time for well-prepared initial data, while the TS-MTI method is uniformly and optimally accurate in time for all kind of initial data. Extensive numerical results are reported to show their efficiency and accuracy, especially in the subsonic limit regime. Finally, the methods are applied to study numerically convergence rates of ZS to its limiting models when ε → 0.