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

Weizhu Bao Department of Mathematics, National University of Singapore, Singapore 119076 Chunmei Su Beijing Computational Science Research Center, Beijing 100193,China, and Department of Mathematics, National University of Singapore, Singapore 119076

Numerical Analysis and Scientific Computing mathscidoc:2205.25015

SIAM Journal on Scientific Computing, 40, (2), A929-A953, 2018.3
We present two uniformly accurate numerical methods for discretizing the Zakharovsystem (ZS) with a dimensionless parameter 0< ε ≤ 1, which is inversely proportional to theacoustic speed. In the subsonic limit regime, i.e., 0< ε << 1, the solution of ZS propagates waves with O(ε)- andO(1)-wavelengths in time and space, respectively, and/or rapid outgoing initial layerswith speed O(1/ε) in space due to the singular perturbation of the wave operator in ZS and/or theincompatibility of the initial data. By adopting an asymptotic consistent formulation of ZS, wepresent a time-splitting exponential wave integrator (TS-EWI) method via applying a time-splittingtechnique and an exponential wave integrator for temporal derivatives in the nonlinear Schr ̈odingerequation and wave-type equation, respectively. By introducing a multiscale decomposition of ZS, wepropose a time-splitting multiscale time integrator (TS-MTI) method. Both methods are explicitand convergent exponentially in space for all kinds of initial data, which is uniformly for ε ∈ (0,1].The TS-EWI method is simpler to be implemented and it is only uniformly and optimally accuratein time for well-prepared initial data, while the TS-MTI method is uniformly and optimally accuratein time for any kind of initial data. Extensive numerical results are reported to show their efficiencyand accuracy, especially in the subsonic limit regime. Finally, the TS-MTI method is applied tostudy numerically convergence rates of ZS to its limiting models when ε→0+.
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@inproceedings{weizhu2018a,
  title={A Uniformly and Optimally Accurate Method for the Zakharov System in the Subsonic Limit Regime},
  author={Weizhu Bao, and Chunmei Su},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220519163505081706294},
  booktitle={SIAM Journal on Scientific Computing},
  volume={40},
  number={2},
  pages={A929-A953},
  year={2018},
}
Weizhu Bao, and Chunmei Su. A Uniformly and Optimally Accurate Method for the Zakharov System in the Subsonic Limit Regime. 2018. Vol. 40. In SIAM Journal on Scientific Computing. pp.A929-A953. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220519163505081706294.
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