Uniform Error Bounds of a Finite Difference Method for the Zakharov System in the Subsonic Limit Regime via an Asymptotic Consistent Formulation

Abstract

We establish uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system (KGZ) with a dimensionless parameter ε ∈ (0, 1], which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e. 0 < ε ≪ 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(ε)-wavelength in time and O(1)-wavelength in space as well as outgoing initial layers in space at speed O(1/ε). 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. By adapting an asymptotic consistent formulation, we propose a uniformly accurate finite difference method and rigorously establish two independent error bounds at O(h 2 + τ 2 /ε) and O(h 2 + τ + ε) with h mesh size and τ time step. Thus we obtain a uniform error bound at O(h 2 + τ ) for 0 < ε ≤ 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 ε-dependent error bounds between the solutions of KGZ and its limiting model when ε → 0 + . Finally, numerical results are reported to confirm our error bounds.

Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes

“…Similar to the properties of the Zakharov system [3,28,32], the solution of the KGS equations (1.10) propagates highly oscillatory waves at wavelength O(ε) and O(1) in time and space, respectively, and rapid outgoing initial layers at speed O(1/ε) in space. This highly temporal oscillatory nature in the solution of the KGS equations (1.10) brings significant numerical difficulties, especially when 0 < ε ≪ 1 [3,28,32]. For example, classical methods may request harsh meshing strategy (or ε-scalability) in order to get 'correct' oscillatory solutions when ε ≪ 1 [8,34].…”

“…An asymptotic consistent formulation. Inspired by the analysis concerning on the convergence between the Zakharov system (ZS) and the limiting cubically Schrödinger equation [28] and the uniform method for solving the ZS in the subsonic limit regime [3], we introduce…”

“…For simplicity of notation, we only present the numerical method for the KGS system on one space dimension, and extensions to higher dimensions are straightforward. Practically, similar to most works for computation of the Zakharov-type equations [3,30], (2.23) is truncated on a bounded domain Ω = (a, b) with homogeneous Dirichlet boundary condition:…”

“…In such case, in order to make sure ψ ε,1 is uniformly bounded for ε ∈ (0, 1], τ has to be taken as τ ε −β/2 , which is too restrictive. To rescue this, we replace ψ 2 (x) above by a modified version [3]…”

“…For example, classical methods may request harsh meshing strategy (or ε-scalability) in order to get 'correct' oscillatory solutions when ε ≪ 1 [8,34]. Recently, we proposed and analyzed uniform accurate finite difference methods for the Zakharov system [3] and Klein-Gordon-Zakharov system [4] in the subsonic limit regime by adopting an asymptotic consistent formulation. The main aim of this paper is to propose and analyze a finite difference method for the KGS equations, which is uniformly accurate in both space and time for 0 < ε ≪ 1.…”