Study Of High-order WENO Finite Volume Methods For Riemann Problem Of Nonstrictly Hyperbolic Conservation Laws

In this thesis,we study on Riemann problem of a kind of 2 x 2 nonstrictly hyperbolic conservation laws arising in elastic theory.We provides the exact solution of this system,and construct high-order numerical schemes to solve it.Firstly we analysis the basic properties of the systems,provides the basic structure of exact solutions under different Riemann initial conditions,and graphically represented in(u-v)plane and(x-t)plane.The nonstrictly hyperbolic of the system may cause nonlinear interactions between waves,which could destruct the classical Riemann solution structure.Now there are two special cases,solutions containing 2-rarefaction waves and solutions containing 2-shock waves.Then we construct two high-order WENO(Weighted essentially non-oscillatory)fi-nite volume methods for the system,one is classical WENO_JS[18]and the other is new WENO_ZQ[39].The different is that WENO_JS convex combines three low order polyno-mials to reach high order approximation,using linear weights and smoothness indicators to construct non-linear weights,so as to be high-order accurate in smooth regions and essentially non-oscillatory near discontinuity.WENO_ZQ convex combines a high order polynomial with two low order polynomials to reach high order approximation,and have more choice of linear weights,we can choose any positive numbers as long as their sum equals one.Finally,We take numerical experiments with thus two numerical schemes to compare with the exact solution.Numerical experiments shows that two WENO schemes both have high-order accurate in smooth regions,and WENO_ZQ with different,linear weights(we sets three types of linear weights in WENO_ZQ)always perform better in L1 error and L? error.The high resolution to the main characteristics of solution and robustness of two WENO near discontinuity.In contrast,WENO_ZQ has better resolution,but it's easy to be unstable in contact discontinuity.