The Research Of Constructing High-Accuracy And High-Resolution Entropy Consistent Schemes Based On Three-Order CWENO Reconstruction

Hyperbolic conservation laws are the important physical model in Fluid Dynamics.Therefore, the research on the accurate solution of numerical method is very important. In recent years, the entropy conservative, entropy stable and entropy consistent schemes based on the basic theory of the conservation laws and the Theorem of the Second Law of Thermodynamics are of great significant in numerical solution for hyperbolic conservation laws. In this thesis, the main ideas and constructing methods for these three schemes are properly discussed. According to it, these three schemes especially for the Burgers equation and the Shallow Water equation are specifically constructed. On this basis, a new scheme of high accuracy and high resolution is constructed by using the three order-CWENO reconstruction and adding the appropriate limiters. Besides, the validity of this new scheme is verified by numerical experiments. The main works are as follows:(1) The construction ideas and features of the entropy conservative, entropy stable and entropy consistent schemes are studied in detail. The entropy conservation scheme is of two order accuracy and there is no dissipation mechanism with it, which leads the oscillations near the discontinuities. The entropy stable schemes improve the entropy conservation schemes by adding the dissipation mechanism to reduce the oscillations. Based on the entropy stable scheme, the entropy consistent scheme has another small dissipation term to meet the requirement of the dissipation quantity in the sense of the physics, so as to eliminate or reduce the oscillations.(2) In order to compute the numerical solution accurately, this thesis constructed a new high accuracy-high resolution scheme by using three order-CWENO reconstruction and adding appropriate limiters on the base of entropy consistent scheme. This scheme not only avoids the non-physical phenomena but also captures the rarefaction waves and shock waves accurately, which makes it great superiority in solving the hyperbolic conservation laws.(3) Verified the validity of the new scheme by the numerical simulations for Burgers equation.(4) According to the constructing process of the Burgers equation, the entropy stable,entropy consistent, high resolution entropy consistent, and high accuracy- high resolutionentropy consistent scheme are extended for shallow water equations. By comparing the results of numerical simulations for each scheme, the advantages of the new high accuracy-high resolution entropy consistent are verified.