一个新的Riemann解法器及相关问题的研究

【作者】 吴昊；

【导师】 沈智军；

【作者基本信息】 中国工程物理研究院 ， 计算数学， 2006， 硕士

【摘要】 本文研究双曲守恒律方程的Godunov型格式。全文分为两部分。第一部分提出了一个新的Riemann解法器，利用两步分裂的思想，对原Riemann问题的求解转化为两个相关的子问题的求解。这种方法可以避开避开传统Riemann解法器中复杂的波系分解和通量选择，较为方便地构造一个原Riemann问题的近似解。将这个新Riemann解法器写成等价的数值通量形式，可以发现该格式是单调的，利用Harten定理证明了该格式是TVD的，并证明了该格式满足离散熵条件。大量的数值试验表明该Riemann解法器的健壮性好，应用范围广泛。 第二部分研究了Burgers方程和Euler方程数值计算中的声速点故障问题。对数值格式产生该现象的原因进行了分析。应用前述的两步分裂的思想分解原方程，通过增加或减少线性对流项使得每步的子问题中都不包括跨声速区，从而避开了声速点故障现象。将此方法实施于一系列典型格式，均取得了令人满意的结果。更多还原

【Abstract】 This paper consists of two parts. In the first part, the Riemann solvers in Godunov-type schemes are considered in the hyperbolic conservation laws systems. Based on the two-step splitting method, a new approximate Riemann solver is presented. In the first step we add a linear advection term to the equations of hyperbolic conservation laws to make the Godunov numerical flux in the interface between cells is computed easily, then the added linear advection term is thrown off in the second step. During the computing procedure, the algorithm does not need iterative technique and characteristic wave decomposition. After the new solver is expressed a form of a new numerical flux, the new scheme is proved monotonic and TVD and satisfies the entropy condition. A lot of numerical results show its robustness and the solver can be applied widely.In the second part, we show reasons that the sonic point glitch in the computation of conservation law happens. A new method is presented to cure the sonic glitch. The method that based on the two-step splitting method is similar to the present solver presented in the first part. Applying this method to many schemes with sonic point glitch, the sonic glitch is cured.更多还原