The Development And Analysis For Discontinuous Galerkin Methods In Lagrangian-Eulerian Framework

The main work of this paper is to develop and analyze two moving mesh discontin-uous Galerkin methods in the Euler-Lagrangian framework for solving time-dependent partial differential equations.One of them is the arbitrary Lagrangian-Euler discontinu-ous discontinuous Galerkin(ALE-DG)method,which can be coupled with an adaptive mesh method to capture the properties of the local solution,and it can also reduce nu-merical dissipation and improve accuracy.Here,we apply the ALE-DG method to the hyperbolic equation involving ?-singularity and Korteweg-de Vries(KdV)equation on moving meshes,and give the stability analysis and error estimates.Another moving mesh discontinuous Galerkin method is to approximate the characteristic line to get a relatively large time-stepping size.We propose a generalized Euler-Lagrangian discon-tinuous Galerkin(GEL-DG)method,and we apply it to the scalar transport equation to obtain extra large time-stepping size.Then we will apply it to system later.The research in this article is mainly divided into three parts.In the first part,we develop and analyze an ALE-DG method for solving one-dimensional hyperbolic equations involving ?-singularities on moving meshes.The L2 and negative norm error estimates are proven for the ALE-DG approximation.More precisely,when choosing the approximation space with piecewise kth degree polyno-mials,the convergence rate in L2-norm for the scheme with the upwind numerical flux is(k+1)th order in the region apart from the singularities,the convergence rate in H-(k+1)norm for the scheme with the monotone fluxes in the whole domain is kth or-der,the convergence rate in H-(k+2)norm for the scheme with the upwind flux in the whole domain can achieve(k+1/2)th order,and the convergence rate in H-(k+1)(R\RT)norm for the scheme with the upwind flux is(2k+1)th order,where RT is the pollution region at time T due to the singularities.Moreover,numerically the(2k+1)th order accuracy for the post-processed solution in the smooth region can be obtained,which is produced by convolving the ALE-DG solution with a suitable kernel consisted of B-splines.Numerical examples are shown to demonstrate the accuracy and capability of the ALE-DG method for the hyperbolic equations involving ?-singularity on moving meshes.In the second part,several ALE-DG methods are presented for KdV type equa-tions on moving meshes.Based on the L2 conservation law of KdV equations,we adopt the conservative and dissipative numerical fluxes for the nonlinear convection and lin-ear dispersive terms respectively,thus one conservative and three dissipative ALE-DG schemes are proposed for the equations.The invariant preserving property for the con-servative scheme and corresponding dissipative properties for the other three dissipative sche1es are all presented and proven in this paper.In addition,the L2 norm error esti-mates are also proven 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 k-th degree polynomials,the error estimate provides the k-th order of convergence rate in L2-norm for the scheme with the conservative numerical fluxes applied for the nonlinear convection term.Furthermore,the(k+1/2)-th order of accu-racy 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.In the third part,we propose a GEL-DG method.The method is a generalization of the Eulerian-Lagrangian discontinuous Galerkin(EL-DG)method for transport prob-lems proposed,which tracks solution along approximations to characteristics in the DG framework,allowing extra large time stepping size with stability.The newly proposed GEL-DG method is motivated for solving linear hyperbolic systems with variable co-efficients,where the velocity field for adjoint problems of the test functions is frozen to constant.In a simplified scalar setting,we propose the GEL-DG methodology by freezing the velocity field of adjoint problems,and by formulating the semi-discrete scheme over the space-time region partitioned by linear lines approximating character-istics.The fully-discrete schemes are obtained by method-of-lines Runge-Kutta(RK)methods.We further design flux limiters for the schemes to satisfy the discrete geo-metric conservation law(DGCL)and maximum principle preserving(MPP)properties.Numerical results on 1D and 2D linear transport problems are presented to demonstrate great properties of the GEL-DG method.These include the high order spatial and tem-poral accuracy,stability with extra large time stepping size,and satisfaction of DGCL and MPP properties.