A Study Of High Order Numerical Methods Of Hyperbolic Conservation Laws And Related Equations

Generally speaking,many physical phenomena can be described by nonlinear partial differential equations(PDE),which as time evolves,may develop singular solutions,such as shock waves,detonation waves,etc.If the global uniform mesh method 1s used,the computation cost will be huge even impossible,when we would like to capture the fine structures using numerical methods,especially for two or higher dimensional systems.In this case,the adaptive mesh methods that can increase the accuracy of the numerical approximations and decrease the storage cost are in critical need.In this thesis,we have proposed to develop a moving mesh discontinuous Galerkin(DG)method.Besides,the local discontinuous Galerkin(LDG)method,which is an extension of DG method,can effectively solve PDEs with higher than first order spatial derivatives.However,it has the disadvantage of employing a large number of unknown variables and the number increases rapidly as the order of the method increases,especially for multiple dimensions and high order PDEs.To avoid the disadvantages of DG or LDG,we have proposed a hybrid LDG-HWENO scheme for the numerical solution of KdV-type PDEs.This thesis contributes in the following two aspects:1.A moving mesh DG method is developed for the numerical solution of hyper-bolic conservation laws.The method is a combination of the DG method and the mesh movement strategy.The moving mesh strategy is based on the moving mesh partial dif-ferential equation(MMPDE)approach,where the mesh is moving continuously in time and orderly in space.We discretize the hyperbolic conservation laws on moving meshes in the quasi-Lagrangian fashion,with which the mesh movement is treated continuously and no interpolations are needed for physical variables from the old meshes to the new ones.Two convection terms are induced by the mesh movement and their discretizations are incorporated naturally in the DG formulation.Numerical results for a selection of one-and two-dimensional scalar and system conservation laws are presented.It is shown that the moving mesh DG method achieves the desired orders of convergence for problems with smooth solutions and is able to capture shocks with grid points concentrated around shocks.We also show its advantage as comparing to uniform meshes and its insensitive?ness to the smoothness of meshes.2.A hybrid LDG-HWENO scheme is proposed for the numerical solution of KdV-type PDEs.The approach is motivated by the hybrid RKDG-HWENO and PNPM methods.It evolves simultaneously the cell averages of the physical solution and its moments(a feature of Hermite WENO)while discretizes high order spatial derivatives using the local DG method.This new scheme incorporates the advantages of both LDG and HWENO methods.It can deal with high order spatial derivatives,but only use a small number of global unknown variables,which is independent of the order of the scheme and the spatial order of derivatives for the underlying differential equations.We demonstrate our new approach by using a set of one and two dimensional numerical examples,which show that our scheme can obtain the same formal high order accuracy as the original LDG method.