The Research Of High Performance Numerical Methods For Hyperbolic Conservation Laws

Numerical simulation methods are of very important for Computational Fluid Dynamics.A great number of research about numerical methods have been paid to nonlinear hyperbolicconservation laws due to its wide range of applications and being difficult to solve. But theprevious numerical methods would most likely not adhere to physical systems——the secondlaw of thermodynamics, which is entropy stable condition, and have generated somenon-physical phenomena. To solve this problem, this paper based on physical concepts, andpresents the research and development of constructuring the entropy conservation/entropystability/entropy consistent schemes for different hyperbolic conservation laws, such asinviscid Burgers equation, Euler equations, shallow water equations. High resolution entropyconsistent schemes were achieved by means of embedding some suitable limiters innumerical fluxes. A second-order rotated Riemann solver approach was presented for twodimensional Euler equations. The main innovations of this work were as follows:(1) The constructuring procedures of the entropy conservation/entropy stability/entropyconsistent scheme are presented. Entropy conservation ensures that entropy is neither creatednor destroyed, and this is true for smooth flows, and it is second order accuracy in this paper.Entropy stability ensures that the entropy change within the system is of the correct sign, andit is entropy-conserved flux with added Roe viscosity term. A scheme is then said to beentropy consistent if it generates entropy with the correct sign and magnitude across anydiscontinuity such as shock or contact discontinuities. It is an entropy stability flux added asmall viscosity that ensures the scheme’s monotonicity, so the numerical diffusion part willaccurately produce the proper amount of entropy, consistent with the second law ofthermodynamics. High resolution entropy consistent scheme was achieved by using suitablelimiters embedding in entropy consistent fluxes.(2) The entropy consistent scheme for one-dimensional inviscid Burgers equation wasstudied, specifically looking at the shock results of entropy consistent flux combined withlimiter. The approach was then repeated for one-dimensional Euler equations. Overall, the numerical experiments demonstrated that the new scheme achieved high resolution whenusing suitable limiters.(3) A second-order Euler flux function based on a rotated Riemann solver approach waspresented in this paper. This scheme is different from the grid-aligned finite-volume methodor the finite difference method based on dimensional splitting. It is based on a rotatedRiemann solver approach and is developed by a particular combination of the HLLC solverand the HLL solver. The HLL solver is applied in the direction normal to shocks to suppresscarbuncles and the HLLC solver is applied across shear layers to avoid an excessive amountof dissipation. The new rotated-hybrid scheme is extremely simple, carbuncle-free and highlyefficient.(4) Based on the experience of solving Burgers and Euler equations, entropyconservation and entropy stability fluxes are developed for the shallow water equations. Tooffset the entropy production of the cube order of the shock strength across the shock waves,the amount of absolute value characteristic velocity difference were added to the entropystable flux, so as to achieve a new numerical scheme similar to Euler equations’s entropyconsistent scheme. The new numerical difference scheme is extremely simple, high efficiency,without adding any artificial numerical viscous terms. Numerical experiments of the proposedscheme adequately demonstrate these advantages. The new scheme successfully simulates thecircular shock water wave propagations with the different kind of dam break problem, so thisis a more ideal method for solving the shallow water equations. But numerical experimentsalso show that the new scheme produces more dissipation for total entropy, so it is just a kindof entropy stability scheme, not a real entropy consistent scheme.