The Research Of Adaptive Artificial Viscosity Entropy Stable Scheme For Hyperbolic Conservation Laws

In recent years,as one of the important topics in the study of computational fluid mechanics,the numerical solution of hyperbolic conservation laws has been paid much attention.Based on the theory of entropy conservation laws and the Theorem of the Second Law of Thermodynamics,three basic entropy schemes are developed: entropy conservative,entropy stable and entropy consistent schemes.The entropy conservation scheme is of two order accuracy and there is no dissipation mechanism with it,which leads the oscillation near the discontinuities.The entropy stable scheme is based on the entropy conservation scheme,adding the dissipation term,reducing the oscillation and achieving the effect of entropy stable,but the inaccurate amount of dissipation leads to the numerical smearing phenomenon in the calculation process.Adding viscosity item from Ismail based on the entropy stable scheme,entropy consistent scheme meets the requirement of the cube order of the shock strength when solution across the discontinuity of shock,so as to eliminate the oscillations.Currently,entropy consistent scheme is the most accurate numerical scheme for estimating the change of entropy,but it is complicated to prove whether a scheme is entropy consistent.Therefore,we are left with nothing better than the second choice,researching the entropy stable method,which is high accuracy by adding adaptive artificial viscosity on entropy conservative scheme.The main work of this paper includes:(1)A new entropy stable scheme,adaptive artificial viscosity entropy stable scheme,is constructed by using entropy conservative scheme and adaptive artificial viscosity.It not only can inhibit oscillations very well,but also improve numerical smearing phenomenon,and has better accuracy when solving hyperbolic conservation laws.The adaptive adaptation of artificial viscosity is achieved by the weak local residual WLR,the viscous coefficient C and the WLR in the artificial viscosity are also given in detail in this paper.(2)The adaptive artificial viscosity entropy stable schemes are structured to solve the Burgers equations,by comparing the results of numerical simulations for entropy conservative,entropy stable and entropy consistent scheme,the advantages of new scheme are verified for Burgers equations.(3)The adaptive artificial viscosity entropy stable scheme is used to solve the shallow water equations.Numerical tests for the shallow water equations also verified the features of the new schemes,such as non oscillation,high precision,high resolution,and robustness.