The Research Of High Performance Numerical Methods For Hyperbolic Conservation Laws

It is one of the important research contents in computational fluid dynamics for solving nonlinear hyperbolic conservation laws. The detailed construction procedures of entropy conservation / entropy stable / entropy consistent schemes for solving inviscid Burgers equation and Euler equations were introduced. A class of high resolution entropy consistent schemes were obtained by combining appropriate limiters and ENO/WENO reconstruction methods in the entropy consistent schemes for solving the equations above. Several upwind flux splitting schemes were introduced, and the analysis of their shock stability was given. The main innovations of this work were as follows:(1) The basic theories and construction procedures of entropy conservation / entropy stable/ entropy consistent schemes were given, and inserted flux limiters and WENO reconstruction methods into entropy consistent schemes in order to get the high resolution characteristic feature. The reconstruction procedures of entropy consistent schemes with limiters and WENO entropy consistent schemes were discussed in detail. Finally, the schemes were applied to the numerical examples of one-dimensional inviscid Burgers equation and Euler equations to verify the good characteristics.(2) Several upwind flux splitting schemes were introduce in detail, and the analysis of their shock wave stability was given. Liou’s splitting scheme, Zha’s splitting scheme, TV’s splitting scheme, HLL-CPS method and the HLL-CPS-T method which based on TV splitting scheme and HLL-CPS method were described contrastively, the low dissipation and shock stability of the HLL-CPS-T method were proved. The conclusion: a sufficient condition for upwind method to ensure shock stability is found to be that: velocity perturbation is attenuated, and the theoretical analysis to verify the correctness of the conclusion was given. Finally, the two-dimensional numerical examples, the inviscid supersonic flow around a circular cylinder, and shock diffraction over a 90 o corner were chosen to verify the good characteristics of HLL-CPS-T method which proposed above.