The Research Of Entropy Stable Schemes For Hyperbolic Conservation Laws

Numerical methods for nonlinear hyperbolic conservation laws are one of important topics in computational fluid dynamics. Since the solutions often contain discontinuities, many numerical methods do not guarantee that the solutions are physical. To solve this problem, a new method called entropy stable scheme is studied and developed. This method is related closely to physical concepts, equips with sufficient theoretical basis, requires no artificial parameters but also avoids the unphysical phenomena. All these virtues show a good application prospect of the method in solving hyperbolic conservation laws.To improve the accuracy of the existing entropy stable schemes, high resolution methods those were rapidly developed recently are introduced. By inserting symmetrical limiters and using high order reconstructions at cell interfaces, a new high resolution entropy stable scheme is acquired. The new scheme satisfies the entropy stable condition while captures sharply at discontinuities like shocks. Most numerical experiments in the thesis are tested the first time with this method. Numerical results show the virtues of the present scheme: versatile, robust, accurate and essentially non-oscillations. The main achievements of work are as follows:(1) The design process and related theories of entropy conservative/entropy stable schemes are systematically described. Entropy conservative scheme that maintains total entropy conservative is constructed and proved to be second order accuracy. Based on the comparison principle, a series of common three-point schemes are discussed and relevant conclusions are verified by numerical experiments. Then, combine the entropy conservative scheme and Roe scheme to construct a class of entropy stable scheme named ERoe scheme. ERoe scheme keeps good capture results at discontinuities.(2) One class of high resolution entropy stable scheme is constructed. Symmetrical limiters are introduced to achieve the self-adaption:high accuracy in smooth regions and sufficient viscosities to avoid oscillations at shocks. Then, by using WENO method as reconstruction at cell interfaces, entropy stable schemes are extended to higher orders. Finally, extend the entropy conservative/entropy stable schemes to two-dimensional scalar cases, numerical results verify that the characteristics are still maintained.(3) Extend the entropy conservative/entropy stable schemes to hyperbolic conservation systems. Explicit pathwise entropy conservative scheme for general systems is described. For Euler equations, the paths in phase space are computed by using the eigensystem of the Roe's approximate matrix. For two dimensional shallow water equations, a relatively simple entropy conservative scheme is adopted to improve the computational efficiency. Then, by inserting limiters and using WENO method, high resolution entropy stable schemes for these two equations are explored. A series of numerical experiments demonstrate that the entropy conservative/entropy stable schemes accurately capture the structure of solutions.