A Least Square Finite Volume Method For 3D Hyperbolic Conservation Laws On Unstructured Meshes

Hyperbolic conservation laws is extensively used in hydromechanics,aerodynamics, Aircraft Industries, biology and many other engineering areas. It'scommonly that there is no analysis solution for it, which inspires the deeper study anddevelopment of some kinds of numerical method, like finite difference method (FDM),finiteelementmethod(FEM),andfinitevolumemethod(FVM).Theprimaryadvantages ofFVMarenumerical robustness,applicabilityongeneralunstructured meshes, and the intrinsic local conservation properties of the resultingschemes. So FVM are a class of discretization schemes that have proven highlysuccessful in approximating the solution of a wide varietyof conservation law systems,some examples include computational fluid mechanics, meteorology, computationalelectromagnetics,semi-conductordevicesimulation,andcomputationalbiology.A key tool in the design and analysis of finite volume schemes suitable fordiscontinuity capturing is discrete maximum principle analysis. The self-adaptivetechniques of meshes and arithmetic have attracted much attention in hyperbolicconservation laws'equations, which solution has the character of wave and osculationgap which means it changes rapidly only in local and narrow regions. We construct aclass of non-oscillatory finite volume method for 3-D Euler equations on unstructuredtetrahedralmeshes.Thefollowingworkhasbeendoneinthisthesis:1. This article reviews elements of the foundation and analysis of finite volumemethods for 3D conservation law systems, and it also describes the mesh and simplex,TVDRunge-Kuttatimediscretizationmethod,fluxfunctionetc.2. We construct a class of non-oscillatory finite volume method for 3-D Eulerequations on unstructured tetrahedral meshes. We get the linear interpolation on everytetrahedral simplex by using the least square method and ensure that the solution willsatisfy with the maximum principle by using the slope limiter. We choose Roe'sRiemannSolver(orHLLC)as flux function,and useTVDRunge-Kuttamethodintimediscretization. The method avoids choosingtemplate, needs less computation than ENOmethodandhashighresolutiononallcomputationalfield.3. We present a moving mesh method based on spring principium and apply it insolving 3-D Euler equations succefully. We give the updated map from old meshes tonew meshes by using a weighted reconstruction method which satisfies conservationlaws.