Arbitrary Lagrangian-Eulerian Discontinuous Galerkin Methods for KdV Type Equations

In this paper, several arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods are presented for Korteweg-de Vries (KdV) type equations on moving meshes. Based on the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} conservation law of KdV equations, we adopt the conservative and dissipative numerical fluxes for the nonlinear convection and linear dispersive terms, respectively. Thus, one conservative and three dissipative ALE-DG schemes are proposed for the equations. The invariant preserving property for the conservative scheme and the corresponding dissipative properties for the other three dissipative schemes are all presented and proved in this paper. In addition, the L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document} -norm error estimates are also proved for two schemes, whose numerical fluxes for the linear dispersive term are both dissipative type. More precisely, when choosing the approximation space with the piecewise kth degree polynomials, the error estimate provides the kth order of convergence rate in L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-norm for the scheme with the conservative numerical fluxes applied for the nonlinear convection term. Furthermore, the (k+1/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(k+1/2)$$\end{document}th order of accuracy can be proved for the ALE-DG scheme with dissipative numerical fluxes applied for the convection term. Moreover, a Hamiltonian conservative ALE-DG scheme is also presented based on the conservation of the Hamiltonian for KdV equations. Numerical examples are shown to demonstrate the accuracy and capability of the moving mesh ALE-DG methods and compare with stationary DG methods.