The Research Of Entropy Consistent Schemes 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,which is entropy stable condition, and have generated some non-physical phenomena. Basedon physical concepts, a new method called entropy consistent scheme which was entropystable, accurate and high resolution was studied and applied to various equations. The mainachievements of this work were as follows:(1) Firstly, the basic theory of hyperbolic conservation laws was introduced and someimportant definitions such as weak solution, entropy pair, entropy inequality were given.Later, a brief introduction of the finite volume method was given. Finally, a detaileddiscussion about ENO reconstruction and WENO reconstruction was presented to lay thefoundation for the improvements of the existing schemes.(2) 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.(3)2p-th order accurate entropy conservative scheme could be obtained with linearcombinations of second-order entropy conservative fluxes in different stencils. Somedissipative mechanism should be added to ensure entropy stable condition. Arbitrarily highorder accurate entropy consistent scheme was constructed with combination of high orderentropy conservative flux and ENO reconstruction procedure. However, ENO reconstructionhas many shortcomings. In this paper, Weighted ENO reconstruction was used to substituteENO reconstruction. The new scheme was applied to a variety of examples. Numericalresults showed that our scheme was robust, accurate, less time-consuming and simple inboundary treatment.