Error estimations of the fourth-order explicit Richardson extrapolation method for two-dimensional nonlinear coupled wave equations

Although explicit finite difference method (EFDM) for linear wave equation is conditional stability, it has many advantages of low computational memory, cheap computational cost and easy implementation. Meanwhile, stable criterion is comparatively good and acceptable. In this article, an EFDM is generalized to solve nonlinear coupled wave equations in two spaces. Using the discrete energy method and introducing new analytical techniques, it is strictly shown that their solutions are conditionally convergent with an order of O(τ2+hx2+hy2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(\tau ^{2}+h_{x}^{2}+h_{y}^{2})$$\end{document} in H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{1}$$\end{document}- and L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }$$\end{document}-norms as the nonlinear terms and their derivatives are locally bounded, instead of global and uniform boundedness, and the nonlinear terms satisfy local Lipschitz condition. Besides, a Richardson extrapolation method (REM) is developed to provide the approximate solutions with a convergent order of O(τ4+hx4+hy4)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {O}(\tau ^{4}+h_{x}^{4}+h_{y}^{4})$$\end{document} in H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^{1}$$\end{document}- and L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{\infty }$$\end{document}-norms. It it worth mentioning that the proposed REM has the same stable condition as original EFDM, thus further improving computational efficiency. Finally, numerical results verify the highly computational efficiency of the algorithms.