Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations

Abstract

<p style='text-indent:20px;'>In this paper, we investigate an accurate and efficient method for nonlinear Maxwell's equation. DG method and Crank-Nicolson scheme are employed for spatial and time discretization, respectively. A semi-explicit extrapolation technique is adopted for the linearization of the nonlinear term. Since the proposed scheme is semi-implicit, only a linear system needs to be solved at each time step. Optimal convergent order of <inline-formula><tex-math id="M1">\begin{document}$ O(\tau^2+h^{p+\frac{1}{2}}) $\end{document}</tex-math></inline-formula> is proved under time step size condition <inline-formula><tex-math id="M2">\begin{document}$ \tau\leq h^{d/4} $\end{document}</tex-math></inline-formula>. Finally, 2D and 3D numerical examples are provided to validate the theoretical convergence rate.