The Research Of Constructing High-resolution Entropy Consistent Schemes Based On Flux Limiters

The importance of the studying of the numerical methods for hyperbolic conservation laws whichoften represent some Physical models in Fluid Dynamics can not be ignored. In recent years, the entropyconservative, entropy stable and entropy consistent schemes based on the theory of entropy conservationlaws which represent the second law of thermodynamics directly are of great significant for the equationsof hyperbolic conservation laws. In this paper, some related theory, ideas and constructed methods for thesethree classes schemes were properly discussed, and on this basis, we construct new high-resolutionschemes for both scalar and system. Details are as follows:(1) Firstly, the development course of three schemes-entropy conservative, stable and consistent werebriefly introduced and two important concepts-weak solutions and entropy stable condition in the theoriesof Computational Fluid Dynamics were defined in order to give readers a study background and laid afoundation for next parts of paper.(2) Based on the entropy conservative theory of Tadmor, the definition of entropy production wasintroduced. This definition is a mirror of comparison principle presented by Tadmor, and it goes todistinguish the meaning of three schemes above quantitatively by means of mathematical expression. Theentropy production of entropy conservative schemes will vanish; the quantity of entropy production to meetstable which only ensure the right direction of entropy change is not sufficient to counteract oscillationscompletely, although it is less than zero; while the entropy consistent not only reached the stable, but canmeet the requirement of the cube order of the shock strength when solution across the discontinuity ofshock, which result in that the entropy consistent schemes have enough entropy dissipation to removeoscillations near discontinuity. Under these analyses, several constructive methods of entropy conservativeschemes, stable schemes based on upwind numerical dissipation and consistent schemes recommended byIsmail were introduced. The numerical experiments of these schemes were conducted, whose resultsdemonstrated their own characteristics and the progressive relationship of them to capture discontinuitysolutions.(3) In fact, the entropy consistent schemes have only one order accuracy because of the disturbanceof upwind dissipative term. This is also the reason of excessive smearing in discontinuity. To improve theiraccuracy, this paper proposed the high resolution entropy consistent schemes by inserting a new flux limiterinto entropy consistent schemes. This approach is based on the traditional thought of using limiter mechanism to construct high resolution schemes. Finally, the results of numerical experiments show thatnew schemes indeed improve the resolution of discontinuity.(4) To construct the high resolution entropy consistent schemes, the commonly used limiters-upwindand symmetrical were discussed in the paper. We can obtain the conclusions that the high resolutionentropy stable schemes of upwind limiters are sharper in the discontinuous parts of the computed solutionthan those symmetrical counterparts. Based on these studying, we extend Roe’s Superbee limiter to a classof limiters which are named after GSbee by adding some variable parameters. And a new S-M limiter wasconstructed under the inspiration of the regularity of these parameters. This limiter appears the fine featuresin constructing the high resolution entropy stable schemes. At the end, several numerical experiments givean explanation to these characteristics from the visual. And the new S-M limiter was chose to construct thehigh resolution entropy consistent schemes.(5) Finally, the entropy consistent schemes and corresponding high resolution schemes ofone-dimensional were simply extended to two-dimensional problems, and the results of several numericalexamples also show the applicability and superiority of new high resolution schemes.