多流体无散度单压力模型初值问题的渐近性研究
Asymptotic study of the initial value problem to a standard one pressure model of multifluid flows in nondivergence form
摘要：
我们构造家庭的近似解的初值问题，并提供完整的数学证明，他们往往满足标准系统的等温一个压力两流体流动的1-D时，数据是$L^1$的密度和$L^\infty$的速度。为此，我们使用的方法，减少这个系统的偏微分方程的一个家庭的系统的四个常微分方程在巴拿赫空间中的光滑的解决方案是这些近似解。这种方法是有建设性的：使用常微分方程的标准数值方法，人们可以容易地和精确地计算这些近似解，因此，从数学证明，这些近似解可以用于与数值方案进行比较。一个观察协议与以前已知的解决方案从科学计算[S. Evje，T.变平。通用双流体模型的混合通量分裂格式。J. Comput. Physics 192，2003，第175-210页]。我们表明，一个恢复这些作者的解决方案（正是在一种情况下，在另一种情况下略有不同）。然后，我们提出了一个有效的数值格式的两个流体流动的原始系统，并显示它给出了完全相同的结果，上述理论解。
英文摘要：
We construct families of approximate solutions to the initial value problem and provide complete mathematical proofs that they tend to satisfy the standard system of isothermal one pressure two-fluid flows in 1-D when the data are $L^1$ in densities and $L^\infty$ in velocities. To this end, we use a method that reduces this system of PDEs to a family of systems of four ODEs in Banach spaces whose smooth solutions are these approximate solutions. This method is constructive: using standard numerical methods for ODEs one can easily and accurately compute these approximate solutions which, therefore, from the mathematical proof, can serve for comparison with numerical schemes. One observes agreement with previously known solutions from scientific computing [S. Evje, T. Flatten. Hybrid Flux-splitting Schemes for a common two fluid model. J. Comput. Physics 192, 2003, p. 175-210]. We show that one recovers the solutions of these authors (exactly in one case, with a slight difference in another case). Then we propose an efficient numerical scheme for the original system of two-fluid flows and show it gives back exactly the same results as the theoretical solutions obtained above.
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