Study And Application Of Characteristic-Based High Resolution Schemes

This dissertation introduces two types of characteristic-based high order numerical methods for solutions of time-dependent nonlinear partial differential equations, and thus it is composed of two main parts.First,a new finite difference method,which combines Constrained Interpolation Profile(CIP) method and the High-order compact (HOC) method,is proposed to numerically solve the wave propagation problems of nonlinear evolution equations.Second,we discuss a characteristic-based finite volume method for hyperbolic conservation laws,in particular we use it to solve the Euler equations and shallow water equations.In the first part of dissertation,a non-conservative semi-Lagrangian scheme,the CIP-HOC coupling method,is presented for solving some nonlinear waves equations. The main idea of the proposed scheme is as follows.A third-order polynomial is constructed through interpolating the points and derivative values at two adjacent grid points.Unlike to the traditional CIP method,the derivative used in the presented scheme is obtained by using a high-order compact scheme originally proposed by Lele. The position of grid point along the characteristic curves can be obtained by using the semi-Lagrangian method.The evolution of the point value at the next time step is defined by using the obtained polynomial at the position of the grid point.To test the high accurate property of the scheme,we applied to solve the Burger's and KdV equations. A series of numerical experiments are given,and numerical results also verify the effectiveness of the new scheme.In the end,this scheme is also extended to solve one dimensional Euler equations.In the second part,we discuss a characteristic-based finite volume method for hyperbolic conservation laws.We focus on the Euler system due to its importance in gas dynamics.In the scheme,the spatial derivatives are discritized by a finite volume method,while in time,the Simpson's quadrature rule is used.The point values at the cell boundaries are obtained by using the Central WENO reconstruction.The method is composed of the following steps:the position of the grid points along the characteristic curves are computed by using the third or fourth order Runge-Kutta method,while the polynomial is obtained by using the third order or fifth order CWENO reconstruction. Then,we obtain the grid point values by using the polynomial function at the point position.Finally,the numerical flux in the new finite volume scheme is obtained based on the point value.The new high resolution finite volume method is combined with the high resolution property of the characteristic method and the non-oscillatory property of the CWENO method.Some classical tests for both scalar and Euler conservations laws in one dimension are performed to verify he accuracy and convergence of the present scheme.In the end,some benchmark tests of shallow wave equations are also adopted to verify the presented finite volume scheme,such as 1D dam-break with large depth difference,left critical rarefaction,two rarefactions and nearly dry bed,dam-break problem with dry bed and the generation of dry bed in the middle.Furthermore,we also extend the scheme to 2D conservation laws by using the dimensional splitting approach.For the 2D dam-break problems,the performance of the present scheme has been compared with the WAF scheme and other high-order schemes.