变人工黏性的一维van der Waals流体模型稳态解的分析与相关计算

气液流体相变模型解的性态复杂，通常引进常系数的人工黏性以便于分析与计算。对人工黏性系数为比容函数的一维van der Waals等温流体相变在Lagrange坐标下的周期初边值问题进行了研究，分析了其稳态解的性态，给出了只存在平凡解以及存在非平凡解且易于判定的充分条件，并进行了相关计算，同时也对初边值问题进行了计算。计算结果表明，依赖于比容的人工黏性系数抑制相变区解震荡的作用与常数人工黏性系数类似。

Analysis and calculation of steady state solution for a one-dimensional van der Waals fluid model with variable artificial viscosity

Solutions to a model of a fluid with a gas-liquid phase transition are complicated and an artificial viscosity with constant coefficient usually needs to be added to the model to make analysis and computation easier. In this paper, the periodic initial and boundary value problems to a mathematical model in Lagrange coordinates of a 1-D van der Waals fluid with an isothermal phase transition and an artificial viscosity coefficient dependent on specific volume are studied. The behavior of stationary solutions to the model is analyzed and some sufficient conditions, which are easy for use, are given for cases when there are only trivial solutions as well as for cases with non-trivial solutions. Numerical solutions of the initial and boundary value problems are also given. The results show that using an artificial viscosity coefficient dependent on specific volume is of similar efficiency to a conventional artificial viscosity with constant coefficient in terms of controlling oscillations of the solutions in the phase transition domain.