The Application Of High Precision DG Methods For Conservation Laws

Hyperbolic equations provided a powerful model for many phenomena in the fieldof science. Research on computational fluid dynamics, atmospheric physics andaerospace of the frontier projects has shown that hyperbolic equations play a significantrole in dealing with the models. There are a great deal of numerical methods for thehyperbolic conservation laws, such as finite volume method, finite difference and finiteelement method. Local solutions of hyperbolic conservation laws with strongly depend-ing on initial condition, if the initial solution is discontinuous, mainly due to theexistence of its features and the relationship of characteristics, keeping its property.However due to the nonlinear of equation even if the solution is smooth, it occursdiscontinuous solution. As a result it causes trouble on solving solution and affects itsprecision and resolution. Discontinuous finite element method developed in this paperkeeps the advantages while overcoming their shortage of the finite element method andfinite volume method, it doesn’t need to increase the number of meshed nodes, easier toimplement flexibly on the ability to deal with discontinuous solution. Discontinuousfinite element method improves the accuracy depending on the choice of basisfunctions.A reconstruction method for hyperbolic conservation laws on the triangular mesh isdeveloped in this thesis, it could be called point-wise hierarchical reconstruction, whichis based on discontinuous finite element method and finite volume method. In order tocarry out the analysis and verification of reconstruction method better, the wholestructure of the article is as follows:First of all, it introduces the topic purpose, the research goal and the significance,discusses the development process of the discontinuous finite element of at the sametime compared simplely with other numerical solution, showing discontinuous finiteelement appeared in solving practical issue with discontinuities is more significant.Secondly, after spatial discretization and time discretization of the partial differentialequations, it can be turned into ordinary differential equations, and then converted thesolving process with Runge-Kutta discontinuous finite element method, this paperexpounds the new scheme with constraint condition of conservation. Thirdly, the thirdorder and fourth order accuracy of point-wise hierarchical reconstruction format isintroduced in detail, which comes closely to the numerical solution of originalconservation equations by calculating the polynomial and remainder on neighboringcells to obtain the approximate real solution with new coefficient, thereby extends moreeasily to arbitrary grid. Finally, using scalar Burgers equation and the Euler equationwith discontinuous solutions to verify the methods constructed received good results, makes the calculation error is small so that ensures to achieve the required accuracy. Atthe end of this article, summarize the whole paper and discuss the future research workpreliminarily.