The Research Of Entropy Stable/Entropy Consistent Schemes For Hyperbolic Conservation Laws

Computational fluid Dynamics is an interdisciplinary science based on theoretical fluidmechanics、computational mathematics and computer science. With the rapid development ofthe computer technology and numerical computation method, the ability of solving flowproblems with computational fluid dynamics has been improved and numerical simulationsbecome one of the significant means which is comparable to the theoretical research and theexperiment research. Therefore numerical simulation methods are of great significance forComputational Fluid Dynamics. Much attention has been paid to nonlinear hyperbolicconservation laws due to its wide range of applications and being difficult to solve. Theprevious numerical methods tend to ignore an important factor of physical systems——thesecond law of thermodynamics, which is entropy stable condition, and have generated somenon-physical phenomena. To solve this problem, a new method called entropy consistentscheme is studied and developed. This method is related closely to physical concepts, equipswith sufficient theoretical basis, requires no artificial parameters but also avoids theunphysical phenomena. All these virtues show a good application prospect of the method insolving hyperbolic conservation laws.To improve the accuracy of the existing entropy consistent schemes, using entropyconsistent flux combination, a new high resolution entropy consistent scheme is proposed.The new scheme satisfies the entropy stable condition while captures sharply discontinuitieslike shocks. Numerical results show the virtues of the present scheme: reliable, robust,accurate and essentially non-oscillation. The main achievements of work are as follows:(1) Firstly, ENO reconstruction and a sign property were introduced. As the sign propertywas satisfied by ENO based on numerical diffusion operators of arbitrary order of accuracy.So a new high resolution entropy stable ENO scheme is presented through ENOreconstruction. Numerical results showed that entropy stable ENO scheme keeps essentiallynon-oscillatory at discontinuities.(2) Compared with entropy stable scheme, entropy consistent scheme controls entropyproduction more exactly and effectively eliminate phenomena such as expansion shock and spurious oscillations. To improve the accuracy of the existing entropy consistent schemes,using the method of construction of high-order entropy conservation and combining entropyconsistent flux as a new high resolution entropy consistent scheme is presented. Numericalresults showed that our new scheme was robust, accurate, easy to programming and simple inboundary treatment.