The Research Of Entropy Consistent Schemes For Hyperbolic Conservation Laws

Many researchers have paid more attention to numerical methods for nonlinearhyperbolic conservation laws, which is one of the important research topics in computationalfluid dynamics. The previous numerical methods tend to ignore an important factor ofphysical systems–the second law of thermodynamics, which is the entropy stable condition,and have lead to some non-physical phenomena. To solve this problem, a new method calledentropy-consistent scheme is studied. This method is related closely to physical concepts andavoids the unphysical phenomena, such as rarefaction shock and negative pressure. All thesevirtues show a good application prospect in solving hyperbolic conservation laws.To improve the accuracy of the entropy consistent scheme, high resolution methodsthose were rapidly developed recently are introduced. The new scheme by using high orderreconstruction at cell interfaces can not only have high accuracy, but also catch the shockwave exactly. Numerical examples using the new scheme show good performance. The mainachievements of work are as follows:(1) An entropy-consistent flux for the hyperbolic conservation laws is modified. Topreserve entropy stability and offset the entropy production of the order of cube of the shockstrength across the shock waves, a second-order upwind term and a third-order differenceterm are added to the entropy-conservative flux, so as to achieve entropy consistency. Thenew flux exactly preserves the stationary contact discontinuity and does not capture theunphysical rarefaction shock and negative pressure. By using WENO reconstruction at cellinterfaces, a high order accurate entropy consistent scheme is adopted. Several differentnumerical examples were implemented with the entropy-consistent scheme. The resultsshowed that the improved scheme is stable, high resolution and non-oscillation.(2) The original entropy-stable and entropy-consistent schemes were extended to viscousBurgers equation. Numerical examples show that the entropy-stable and entropy-consistentschemes adding physical viscous term both can produce correct entropy with proper viscouscoefficients and can capture the shock wave exactly.