The Research Of Entropy-stable Schemes Based On Fifth-order CWENO For Hyperbolic Conservation Laws

The numerical solutions of hyperbolic conservation laws are one of important topics on the computational fluid dynamics.In this paper,the principles and reconstruction of entropy-conservative schemes,entropy-stable schemes and entropy-consistent schemes,which satisfy the entropy stability condition,are introduced in detail.Although the entropy-conservative scheme is second-order,it may lead the pseudo-oscillations in the range of discontinuities for lacks dissipation.An "appropriate" numerical viscous item as dissipation one is added in the entropy-conservative scheme,and then the entropy-stable schemes and entropy-consistent schemes are obtained,which can avoid the non-physical phenomenon,such as "carbuncle phenomenon" and "expansion shock",etc.As the dissipation works well,the first order can be achieved in the spatial direction.Furthermore,a new scheme of high accuracy and high resolution entropy-stable scheme is constructed by using the appropriate limiters and the fifth order central weighted essentially non-oscillatory(CWENO5)reconstruction on the basis of the entropy-consistent scheme.Finally,the new schemes are used to solve the classic examples of the Burgers equations and the Euler equations.The results show the effects of the new scheme which we constructed.The details in this paper are as follows:(1)An appropriate limiter is introduced to entropy-consistent scheme,and CWENO5 is reconstructed in cell interface,then the CWENO5 entropy-stable scheme for the hyperbolic conservation law equations is constructed.The numerical results are obtained by the new schemes for the one dimensional Burgers equation,and the numerical results which obtained by the existing algorithms are analyzed.They verify the new schemes are high accuracy and high resolution entropy-stable scheme,comparing the entropy-conservative/ entropy-stable/ entropy-consistent/ high resolution entropy-stable scheme/ CWENO3 entropy-stable scheme/ CWENO4 entropy-stable scheme.(2)The CWENO5 entropy-stable scheme of two dimensional Burgers equations is constructed,and the numerical results show that the numerical solution of the new scheme is exacter.The new algorithm can not only avert non-physical phenomena,but also accurately capture shock waves and rarefaction.(3)For Euler equations,the fifth order CWENO entropy-stable scheme is constructed.The results of numerical simulations are analyzed with the numerical results of the exact solution and the other schemes.The results show that the new scheme not only has high accuracy and high resolution,but also can avert the pseudo-oscillations effectively than the classical schemes.