High Resolution Finite Difference Methods And Their Applications

Computational fluid dynamics can be used to simulate a large class of complicated flow phenomena appeared in the physical world. Nonlinear hyperbolic conservation laws, served as the basic governing equations for fluid flows, have attracted much attention during the past several decades due to their wide range of applications. Many numerical methods for solving nonlinear hyperbolic conservation laws have been well developed. However, the state is still far from complete. This thesis mainly studies high resolution finite difference methods for hyperbolic conservation laws, and applies such presented methods to various benchmark examples. The main contents and research results can be described as follows:1. By using a high order reconstruction at cell interfaces, a WENO-based entropy stable scheme is developed for nonlinear hyperbolic conservation laws. In the two-dimensional case, this scheme is also generalized by dimension-by-dimension approach. The proposed scheme is used to test extensive model problems which include scalar equations, Euler equations of gas dynamics and shallow water equations in both one- and two-dimension. By comparing the numerical results with the original entropy stable scheme’s results, we can see remarkable improvements of the presented scheme in resolving discontinuities.2. By introducing a switching function matrix, a self adjusting entropy stable scheme is proposed for solving compressible Euler equations. Such switching function, which possesses a property of being close to one at shocks and vanishing away in smooth areas, controls the amount of numerical dissipation in different locations and makes the numerical dissipation added around discontinuities automatically. This is what we mean by ‘self adjusting’. Several numerical examples of one- and two-dimensional Euler equations are presented to demonstrate the good performance of the proposed scheme.3. A third order entropy stable scheme is developed for approximating hyperbolic conservation laws. Firstly, fourth order entropy conservative flux is constructed by a linear combination of two-point entropy conservative flux on different stencils. Then, a third order non-oscillatory reconstruction based on point values is proposed. Such reconstruction satisfies the so-called sign property. We consider a numerical dissipation term using this reconstruction procedure on(characteristic) entropy variables. Third order entropy stable scheme is obtained by adding third order numerical dissipation term to fourth order entropy conservative flux. The extension to two-dimensional case is straightforward. Finally, a large number of test cases are provided to verify the scheme’s accuracy and validity. Numerical results display that the proposed scheme achieves the expected third order accuracy in both one- and two-dimension and possesses the merits of high resolution and non-oscillation when dealing with discontinuous problems.4. A fourth order semi-discrete central upwind scheme is proposed. Based on Godunov-type central scheme, we compute the Riemann fan’s width by evaluating the local propagation speeds of nonlinear waves and derive a semi-discrete central upwind flux. Combining this flux with Peer’s fourth order non-oscillatory reconstruction, a fourth order semi-discrete central upwind scheme is devised. The resulting scheme shares the simplicity of central schemes, namely, no Riemann solvers are involved. Therefore, it avoids the complicated and time-consuming characteristic decomposition procedure. Numerical simulations are performed on scalar conservation law equations, Euler equations and shallow water equations with bottom topography. The high resolution and non-oscillation properties are verified. Furthermore, the proposed scheme can resolve the complicated and delicate structures of the solution.