Several Numerical Methods For Hyperbolic Conservation Laws

This thesis investigates several numerical methods for hyperbolic conservation laws.The thesis is divided into two parts.The first part extends the Entropy-Ultra-bee scheme developed by Mao and his co-workers for the linear advection equation to nonlinear conservation laws and Euler systems,The second part uses RKDG methods to build a simple numerical method for multi-medium compressible flows.The first contribution of this thesis is to extend the Entropy-Ultra-bee scheme for the linear advection equation to the nonlinear scalar conservation laws,and design a so-called Entropy-Monotone scheme.The author proves that the Entropy-Monotone scheme is TVD and satisfies the entropy condition.Numerical tests show that it is better than the standard Godunov scheme and is almost comparable to the second order ENO scheme.The paper[31]applied the Entropy-Ultra-bee scheme for linear advection equation to the Euler systems.The scheme developed in[31]improves the resolution of the solution in the second characteristic field and overcomes the numerical smearing of contact discontinuities.However,the numerical entropy in that scheme is not computed in a proper way and thus the numerical solutions always suffer from the nonphysical oscillations in the density and entropy.The second contribution of this thesis corrects the problem in[31]and designs a new way to compute the numerical entropy.The new scheme eliminates the nonphysical oscillations in the density and entropy,and is simpler than the original scheme in[31].Numerical tests show that the new scheme is efficient,overcomes the numerical smearing of the contact discontinuities,and improves the resolution of the numerical solution in the second characteristic field.The third contribution of this thesis is try to extend the Entropy-Ultra-bee scheme for the linear advection equation and the Entropy-Monotone scheme for nolinear scalar conservation equation to the Euler systems.Because the second characteristic field is linear,we applied the Entropy-Ultra-bee scheme to this field.Be- cause the first and third characteristic fields are nonlinear,we applied the Entropy-Monotone scheme to these two fields.Numerical tests show that the built scheme is efficient,it overcomes the numerical smearing of contact discontinuities,improves the resolution of the numerical solution in the second characteristic field and improves partly the resolution of the numerical solution in the first and the third characteristic fields.The fourth contribution of this thesis is to use the RKDG method to compute multi-medium compressible flows.The author designs a numerical method which considers fully the local feature of RKDG.So the scheme is every simple,its storage cost and computation cost reduces a lot.