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摘要
We establish uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov (KGZ) system with a dimensionless parameter epsilon is an element of (0, 1], which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., 0 < epsilon << 1, the solution propagates highly oscillatory waves in time and/or rapid outgoing initial layers in space due to the singular perturbation in the Zakharov equation and/or the incompatibility of the initial data. Specifically, the solution propagates waves with O(epsilon)-wavelength in time and O(1)-wavelength in space as well as outgoing initial layers in space at speed 0(1/epsilon). This high oscillation in time and rapid outgoing waves in space of the solution cause significant burdens in designing numerical methods and establishing error estimates for KGZ system. By applying an asymptotic consistent formulation, we propose a uniformly accurate finite difference method and rigorously establish two independent error bounds at O(h(2) +tau(2)/epsilon) and O(h(2) + tau + epsilon) with h mesh size and epsilon time step. Thus we obtain a uniform error bound at O(h(2) + tau) for 0 < epsilon < 1. The main techniques in the analysis include the energy method, cut-off of the nonlinearity to bound the numerical solution, the integral approximation of the oscillatory term, and epsilon-dependent error bounds between the solutions of KGZ system and its limiting model when epsilon -> 0(+). Finally, numerical results are reported to confirm our error bounds.
- 出版日期2018-9
- 单位北京计算科学研究中心
