多介质流的高分辨率Euler方法

HIGH RESOLUTION EULER METHODS FOR MULTI COMPONENT FLOW

在多介质流动问题中，不同介质有不同的状态方程。这使通量成为间断函数，从而没有通量的Ｊａｃｏｂｉ矩阵。而用Ｅｕｌｅｒ坐标系描述的方程组的很多高分辨率格式都要用到Ｊａｃｏｂｉ矩阵及其特征值和特征向量，即要求通量连续可微。因此必须重新处理整个守恒律方程组。对于γ气体问题将γ看作一个新未知量并增加一个守恒方程，从而使整个方程组的通量成为光滑函数，为高分辨率格式的构造铺平了道路。由于真实流动只遵守三个守恒律，多加的一个守恒律虽然对偏微分方程组没有影响，但对差分方程数值解有影响。这一点在数值实验中已有表现。提出了一个方案将这一影响尽量消除。所用格式可完全照搬单介质流动的任何现有格式。

In multicomponent flow, different component has diffe rent equation of states. This makes the flux discontinuous and no Jacobian matri x exists. In schemes of high resolution for Euler equations, the Jacobian matrix and its eigenvalues as while as its eigenvectors are needed, that is, the flux should be continuously differentiable. So the whole systems of conservation laws should be rearranged. For γ gas, γ is regarded as a new unknown and a new con servation equation is added, thus the flux of the new system become continuously differentiable, the obstacles in the way to high resolution schemes are removed . Since true flows only obey three conservation laws, though the additional cons ervation law does not influence the exact solutions of original differential equ ations, it does influence the numerical solutions of difference equations. This is obvious in numerical experiments. The presented method eliminates this influe nce as much as possible. All schemes of single component flow can be directly us ed. Numerical experiments for one dimensional shock tube problem of multicompone nt flow demonstrate that so designed schemes have the same effect as those of on e component flow.