High Accuracy WENO Finite Volume Schemes For Hyperbolic Conservation Law Systems

In this paper, we study the finite volume schemes of high order accurate essentially non-oscillatory shock-capturing to two dimensional initial boundary value problems. A class of essentially non-oscillatory schemes for the numerical simulation of hyperbolic equations and systems has been constructed. A few extensions have been applied to multi-dimensional simulations of compressible flows on regular structured meshes and unstructured meshes. The ENO scheme is composed of a piece wise polynomial reconstruction from the piecewise constant values of the last time step. It is based on an adaptive selection of a suitable stencil for each cell such that strong oscillations near discontinuities are avoided . hi smooth parts of the solution the scheme is of higher order.In first chapter, on rectangular partition of the x-y plane, the finite volume ENO schemes are developed in semi-discrete form, employing high order Runge-Kutta methods for temporal accuracy. We study the error, accurate analysis, the two dimensional reconstruction based on the cell average of the solution at time t and yield a global, piecewise polynomial. On the base of ENO schemes, we study the WENO schemes, especially using the new smoothness measurement in resolving complicated shock and flow structures.In the chapter 3,we generate unstructured meshes, mainly the triangular meshes by Advancing Front Methods and Delaunay Method.In chapter 4 , we construct high order weighted essentially non-oscillatory schemes on two dimensional unstructured meshes in the finite volume formulation of linear polynomials and quadratic polynomials, the computation of smoothness indicator and non-negative weight coefficients of polynomials. In the paper, we choice the triangular meshes as computation mesh avoiding the formulation of polygon.In the chapter 5, we enforce numerical experiments of gas dynamics problems, such as a steady state regular shock reflection and a mach 3 wind tunnel with a step which demonstrate accuracies and robustness of the methods for shock calculations.