[Paper Review] Uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system in the subsonic limit regime

Weizhu Bao, Chunmei Su|arXiv (Cornell University)|Dec 30, 2016

Advanced Mathematical Physics Problems19 references3 citations

TL;DR

This paper proposes a uniformly accurate finite difference method for the Klein-Gordon-Zakharov system in the subsonic limit regime ($\varepsilon \ll 1$), where solutions exhibit highly oscillatory temporal waves and rapid spatial outgoing layers. By using an asymptotic consistent formulation and integral approximation of oscillatory terms, the authors establish uniform error bounds of $O(h^2 + \tau)$ for all $\varepsilon \in (0,1]$, ensuring robust convergence regardless of small $\varepsilon$. The method is validated numerically with second-order spatial and time accuracy.

ABSTRACT

We establish uniform error bounds of a finite difference method for the Klein-Gordon-Zakharov system (KGZ) with a dimensionless parameter $\varepsilon \in (0,1]$, which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e. $0

Motivation & Objective

• To address the numerical challenge posed by highly oscillatory solutions in the Klein-Gordon-Zakharov (KGZ) system when the acoustic speed is large ($\varepsilon \ll 1$).
• To develop a uniformly accurate finite difference method that maintains convergence across all $\varepsilon \in (0,1]$, including the singular limit $\varepsilon \to 0^+$.
• To rigorously establish $\varepsilon$-independent error bounds for the numerical solution in both space and time.
• To confirm the theoretical error bounds through extensive numerical experiments.

Proposed method

• An asymptotic consistent formulation is derived to capture the singular perturbation behavior of the KGZ system in the subsonic limit regime.
• The method employs an integral approximation technique to handle the highly oscillatory terms arising from the $\varepsilon$-dependent wave dynamics.
• A finite difference scheme is constructed based on the reformulated system, ensuring stability and accuracy across all $\varepsilon \in (0,1]$.
• The energy method is applied to derive $\varepsilon$-dependent error estimates between the numerical and exact solutions.
• Cut-off techniques are used to control the nonlinearity and bound the numerical solution in energy norms.
• Error bounds are validated numerically using the exponential integrator split-step spectral method as a reference for 'exact' solutions.

Experimental results

Research questions

• RQ1 Can a finite difference method be uniformly convergent in both space and time for the Klein-Gordon-Zakharov system when $\varepsilon \ll 1$?
• RQ2 What is the optimal error bound for the numerical solution that remains valid uniformly for all $\varepsilon \in (0,1]$?
• RQ3 How does the asymptotic consistent formulation improve the accuracy of the finite difference method in the presence of highly oscillatory waves and rapid outgoing layers?
• RQ4 What is the convergence rate of the method in space and time, and does it degrade in the resonance regime $\tau \sim \varepsilon$?
• RQ5 Are the theoretical error bounds confirmed by numerical experiments with varying $\varepsilon$, $h$, and $\tau$?

Key findings

• The proposed finite difference method achieves a uniform error bound of $O(h^2 + \tau)$ for all $\varepsilon \in (0,1]$, ensuring robust convergence regardless of the size of $\varepsilon$.
• Two independent error bounds are rigorously established: $O(h^2 + \tau^2/\varepsilon)$ and $O(h^2 + \tau + \varepsilon)$, which combine to yield the uniform bound.
• Numerical results confirm second-order convergence in space and second-order convergence in time for $E^\varepsilon$, with a first-order degradation in the resonance regime $\tau \sim \varepsilon$.
• For $N^\varepsilon$, the method shows second-order convergence when $\tau \lesssim \varepsilon$ or $\varepsilon \lesssim \tau^2$, and first-order convergence in the resonance regime.
• The error in $N^\varepsilon$ scales as $O(\tau^2 / \varepsilon)$ in the upper triangle of Table 4.2, confirming the theoretical $\varepsilon$-dependent bound.
• The numerical experiments demonstrate that the theoretical error estimates are sharp, with convergence rates matching the predicted behavior across all tested $\varepsilon$ values.