A UNIFORMLY AND OPTIMALLY ACCURATE METHOD FOR THE ZAKHAROV SYSTEM IN THE SUBSONIC LIMIT REGIME

We present two uniformly accurate numerical methods for discretizing the Zakharov system (ZS) with a dimensionless parameter 0 < epsilon <= 1, which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., 0 < epsilon << 1, the solution of ZS propagates waves with O (epsilon)-and O (1)-wavelengths in time and space, respectively, and/or rapid outgoing initial layers with speed O (1/epsilon) 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 Schrodinger 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 convergent exponentially in space for all kinds of initial data, which is uniformly for epsilon is an element of(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 any kind of initial data. Extensive numerical results are reported to show their e ffi ciency and accuracy, especially in the subsonic limit regime. Finally, the TS-MTI method is applied to study numerically convergence rates of ZS to its limiting models when epsilon -> 0(+).