Uniform error bounds of a nested Picard iterative integrator for the Klein-Gordon-Zakharov system in the subsonic limit regime

Abstract

We propose a nested Picard iterative integrator Fourier pseudo-spectral (NPI-FP) method and establish the uniform error bounds for the Klein-Gordon-Zakharov system (KGZS) with \(\varepsilon \in (0, 1]\) being a small parameter. In the subsonic limit regime (\(0 < \varepsilon \ll 1\)), the solution of KGZS propagates waves with wavelength \(O(\varepsilon )\) in time and amplitude at \(O(\varepsilon ^{\alpha ^\dagger } )\) with \(\alpha ^\dagger =\min \{\alpha ,\beta +1,2\}\), where \(\alpha \) and \(\beta \) describe the incompatibility between the initial data of the KGZS and the limiting equation as \(\varepsilon \rightarrow 0^+\) and satisfy \(\alpha \ge 0\), \(\beta +1\ge 0\). The oscillation in time becomes the main difficulty in constructing numerical schemes and making the corresponding error analysis for KGZS in this regime. In this paper, firstly, in order to overcome the difficulty of controlling nonlinear terms, we transform the KGZS into a system with higher derivative. Using the technique of nested Picard iteration, we construct a new time semi-discretization scheme and obtain the error estimates of semi-discretization with the bounds at \(O(\min \{\tau ,\tau ^2/\varepsilon ^{1-\alpha ^*}\})\) for \(\beta \ge 0\) where \(\alpha ^*=\min \{1,\alpha ,1+\beta \}\) and \(\tau \) is time step. Hence, we get uniformly second-order error bounds at \(O(\tau ^{2})\) when \(\alpha \ge 1\) and \(\beta \ge 0\), and uniformly accurate first-order error estimates for any \(\alpha \ge 0\) and \(\beta \ge 0\). We also give full discretization by Fourier pseudo-spectral method and obtain the error bounds at \(O(h^{\sigma +2}+\min \{\tau ,\tau ^2/\varepsilon ^{1-\alpha ^*}\})\), where h is mesh size and \(\sigma \) depends on the regularity of the solution. Hence, we get uniformly accurate spatial spectral order for any \(\alpha \ge 0\) and \(\beta \ge 0\). Our numerical results support the error estimates. Surprisingly, our numerical results suggest a better error bound at \(O(h^{\sigma +2}+ \varepsilon ^q\tau ^{2})\) for a certain \(q\in \mathbb {R}\).