The Finite Difference Method For Hyperbolic Conservation Laws

The solutions of the Hyperbolic conservation laws might develop discontinuity even if the initial conditions are very smooth. This loss of smoothness present challenge towards the design of the numerical algorithm. Finite difference method is a very important hot research topic in the field of numerical method, different difference method have different form, from linear form to nonlinear form, from instability to stability, from nonconvergent scheme to convergent scheme, the theory of the finite difference method and other numerical method gradually become mature; on the other hand, with the advance of the resolution of the numerical scheme, the technology of the generation of the mesh also improved ,thus, the adapt scheme which is have some thing to do with these two kinds of advance also have been developed and have been widely used, because it not only considers the resolution of the scheme but also considers costs of time. This thesis begin with the introduction of Blowup of the smooth solution of the Hyperbolic conservation laws, which contained weak solutions, Riemann Problem, Lax shock condition and so on, then, it introduces the general conception and theory about the finite difference method(including the tolerance, convergence and stability of the difference scheme),and then, a detail introduction about the evolution of a typical finite difference algorithm has been introduced in chapter 3(including the classical and nonclassical central method), that is Godunov-central scheme ,finally, in chapter 4,we introduce the adapt algorithm, including algorithm adaption and mesh adaption. The central part of my research work is mesh adaption , mainly focusing on the design of a smoothness indicator and the relative theoretical analysis of the scheme. All adapt schemes demand a smoothness indicator to seek out smooth region and rough region to treat them respectively. By analyzing the disadvantage of an existing smoothness indicator, which is very costly, we present a new smoothness indicator called total-variation smoothness indicator, which is very simple and easy to understand, and is cost-effective. Our numerical examples show that is can accurately find out the rough area. The theoretical analysis is another major work , by refining the mesh in the rough region, our mesh-adaption scheme can simulate rough area more accurately, but the mesh should not be refined without any stop, how to determine the times of refining? that is, when should the refining course stopped? Besides, due to the restriction of CFL conditions, time-step will also reduce by half when the space-step reduced by half, this will increase the number of time steps, causing the number of time steps twice than that of before refining, whether our mesh adaption scheme can ensure the total error won't increase after refining? All these questions need to be answered. By the theoretical analysis, we present a method to control the times of refining ,and give out the comparison of the errors before and after refining.The results show that, the total error of the scheme after refining many times will no more than that of the original scheme.