HWENO Scheme With Two Stage Fourth Order Time Discretization Method For Hyperbolic Conservation Laws

In this thesis,we study the modified finite volume HWENO(Hermite Weighted Essentially Non-oscillatory)scheme and two stage fourth-order time discretization method.Then we combine them and apply to hyperbolic conservation laws.Firstly,we construct the modified finite volume HWENO scheme,which has high order and essentially non-oscillation.We modify the first derivative term of the solution to avoid the derivative value will increase abruptly or nearly infinity at the discontinuity point.Then,we reconstruct the solution and the first derivative terms of the solution by HWENO.Moreover,we use a set of stencil in the reconstruction.Then,we provide the reconstruction of two stage fourth order time discretization method,which is based on Lax-Wendroff type solvers and combined them.As a result,a two-stage procedure can be constructed to achieve a fourth order temporal accuracy,rather than using well-developed four stages for Runge-Kutta methods.Finally,we take amount of numerical experiments to verify the validity of the scheme.Apply to hyperbolic conservation laws in one and two dimensional cases and compare with the time discretization method of third order TVD Runge-Kutta.By comparison with the exact solution,the modified finite volume HWENO scheme combine both two time discretization methods all can reach high-order accurate in smooth regions and essentially non-oscillatory near discontinuity.