The Research Of High Performance Numerical Methods For Hyperbolic Conservation Laws

Numerical simulation methods are of great significance for Computational FluidDynamics. Much attention has been paid to nonlinear hyperbolic conservation laws due to itswide range of applications and being difficult to solve. The previous numerical methods tendto ignore an important factor of physical systems—the second law of thermodynamics, whichis entropy stable condition, and have generated some non-physical phenomena. Based onphysical concepts, a new method called entropy consistent scheme which was entropy stable,accurate and high resolution was studied and applied to various equations. The mainachievements of this work were as follows:(1) Compared with entropy stable scheme, entropy consistent scheme controls entropyproduction more exactly and effectively eliminate phenomena such as expansion shock andspurious oscillations. By using high order Weighted ENO reconstruction at cell interfaces, anaccurate entropy consistent scheme was presented. Several numerical experimentsdemonstrated that the scheme was robust, accurate and essentially non-oscillations.(2) An adaptive mesh method for one-dimensional hyperbolic conservation laws waspresented, namely in areas where the numerical solutions are smoother the grid points arecluster and in areas where the numerical solution has steeper profiles the grid points should besparse. This method could effectively eliminate non-physical phenomena, such as expansionshock, negative density and negative pressure etc., several different numerical examplesshowed that adaptive mesh method is of robustness, reliability and no spurious oscillation.(3) Constructor methods and theories of entropy consistent scheme for solvingtwo-dimensional hyperbolic conservation laws were presented. First introduced constructormethods of solving two-dimensional scalar equation, and solving Burgers equation, to obtainbetter results. Secondly, the construction method for solving the two-dimensional Eulerequation was introduced. Finally, several numerical experiments showed that this method canbe good for solving two-dimensional Euler equations, and effectively eliminate thephenomena of ‘carbuncle’.