Research On High Order Accurate Numerical Methods For Fluid Dynamics

The main purpose of this paper is to study high-order accuracy and high resolution numerical methods on fluid dynamics equations. The main achievements of work is as follows:1. In this paper, extending the traditional GDQ method to solve compressible flow problems of which solutions may be discontinuous, we present a new kind of high-order accurate discontinuous GDQ schemes to solve one dimensional nonlinear hyperbolic conservation laws. The basic idea is the following. Firstly, the computational domain is divided into many small pieces of subdomains. Secondly, in each small subdomain, the GDQ method is implemented. And then, at the boundary interface between small subdomains, the numerical fluxes is chosen according to the flow directions. Afterwards, the GDQ scheme is improved and its non-oscillatory properties are explored. Moreover, the scheme is extended to one-dimensional systems, and two-dimensional scalar conservation laws and systems by splitting and unsplitting form. Finally, many numerical experiments are given, and numerical results verify the validation of the method.2. Based on flux splitting and second-order cell-averaged MUSCL-type reconstruction, second-order accurate schemes is constructed for two dimensional nonlinear scalar hyperbolic conservation laws by taking TVD and UNO limiter. Its MmB property is proved. The extension to the two dimensional system of hyperbolic conservation laws is straight-forward by using component-wise manner. Several numerical results of two dimensional Euler equations are given. The high resolution and robustness of the schemes obtained by two limiters are compared and verified.3. The domain division is similar to discontinuous GDQ partition. Cell-averaged conservative variables at small cell are used to reconstruct conservative variables at the small cell interfaces by high-order interpolation. The correction is introduced to prevent oscillations from the high-order approximation. The approximate Riemann solver is used to evaluate the numerical fluxes at the small cell interfaces, and a high-order conservative scheme for one-dimensional hyperbolic conservation laws is obtained by using high-order Runge-Kutta TVD time discretization. Moreover, we prove the MmB property of the scheme under a certain CFL condition, and extend the scheme to one-dimensional systems and multi-dimensional scalar conservation laws and systems. The high resolution and robustness of the schemes obtained by two cell partition are compared and verified by numerical experiments. Furthermore, based on closely relations of Hamilton- Jacobi equations and the conservation laws, we extend high-order accurate cell-averaged method of hyperbolic conservation laws to solve Hamilton-Jacobi equations. A class of difference schemes of high-order accuracy and high resolution of discontinuity derivative is constructed to Hamilton-Jacobi equations. Numerical results show the success of the present scheme.