High Order Numerical Schemes For Conservation Laws And Relative Equations:Superconvergence And Application

The high order numerical scheme for solving conservation laws gains more and more popularity in the field of computational fluid dynamics due to many attractive properties,such as high order accuracy,high efficiency and high resolution.The high order schemes,such as discontinuous Galerkin(DG)scheme and weighted essentially mon-oscillatory(WENO)scheme can well resolve fine scale solution structures with relatively coarser numerical meshes,thus generating high resolution numerical results with great computational efficiency for large scale numerical simulations.Develop-ing efficient and stable high order methods has always been a hot research topic in the computational science community.In this dissertation,we systematically investi-gate the high order numerical schemes for solving hyperbolic conservation laws and relative equations:firstly,we develop a local structure preserving(LSP)high order Lax-Wendroff(LW)DG scheme for high dimensional Hamilton-Jacobi(HJ)equation;secondly,we analyze the superconvergence properties of DG and LDG schemes;fi-nally,we design high order conservative semi-Lagrangian(SL)schemes for solving lin-ear transport equations and with application to plasma physics and global atmospheric modeling.In the first part,we study the high order scheme for HJ equations,arising in various applications,such as optimal control and image processing.We propose a LSP-DG method with compact one-step one-stage LW type time discretization to capture the viscosity solution for the high dimensional HJ equation.By incorporating the a priori knowledge of the exact solution into the choice of local approximating functions in the DG formulation,the LSP-DG scheme has fewer degrees of freedom and enjoys better efficiency than the standard DG scheme.Moreover,combined with LW time discretiztion method,the resulting LW-LSP-DG scheme is more compact,efficient and requires lower storage than the Runge-Kutta DG scheme.A robust WEND limiting procedure is used to ensure the LW-LSP-DG scheme converges to the viscosity solution of the HJ equation.Numerical results are demonstrated to show the efficiency and effectiveness of the proposed LW-LSP-DG scheme in solving the high dimensional HJ equation.In the second part,via classic Fourier approach,we systematically study the super-convergence properties of DG and LDG schemes.Superconvergence analysis is an im-portant component in understanding the long time behavior of the DG and LDG errors for time dependent hyperbolic and parabolic equations.Based on the eigenstructure analysis of the amplification matrix,we are able to explain that the DG and LDG error will not grow significantly over a long period of time.Specifically,the part of error that grows linearly in time comes from the dispersion and dissipation errors of the physi-cally relevant eigenvalues of amplification matrices.Such error is superconvergent of order 2k + 1 and 2k+ 2 for DG and LDG schemes respectively,compared with the regular k + 1 order of accuracy for DG and LDG scheme.Along this line,we analyzethe error for a fully discrete RKDG scheme.We further propose a new LWDG scheme for solving hyperbolic conservation laws.The newly proposed scheme,compared with the classical LWDG,enjoys superconvergent properties.Specifically,by applying a post-precessing procedure,we observe that the order of accuracy for the new LWDG is enhanced from a regular k + 1-th order to a superconvergent 2k + 1-th order.By using Fourier approach,we investigate the eigenstructures of amplification matrices for these two LWDG approaches to explain better performance of the proposed LWDG scheme.In the third part,we design and implement high order SL schemes for the VP equation and spherical transport equations.The main difficulty associated with the VP simulation is the huge computational cost,due to the 'curse of dimensionality'(3D in physical space and 3D in velocity space).Compared with Eulerian type schemes,the SL type schemes can take arbitrary large time step without stability issue,leading to better computational efficiency.In this dissertation,we propose a SL high order hybrid scheme based on Strang-splitting strategy.We use a SLDG(or RKDG scheme with local time stepping)for spatial advection and use a SL finite difference(FD)WENO scheme for velocity acceleration/decelertion.Such hybrid method takes advantages of DG in its flexibility and compactness and WENO in its robustness in resolving fila-mentation solution structures.Satisfactory numerical results are observed by adopting a set of relatively coarse numerical mesh and large time step.The spherical transport equations are of paramount importance in large scale atmospheric numerical modeling.In application,more than one hundred of transport equations need to be solved at the same time(multi-tracer transport).Therefore,developing efficient numerical solvers is demanding.We propose a SLDG scheme on cubed-sphere grid:the SL scheme is advantageous for its efficiency in using extra large time step,while the cubed-sphere ge-ometry is free of polar singularities compared with the polar coordinate and is ideally suited for element based numerical scheme.A collection of benchmarked numerical tests are used to demonstrate reliability and efficiency of the scheme.