Numerical simulation of a two-phase compressible displacement problem is considered in this paper, modeled by a nonlinear system of partial differential equations (PDEs). Considering the characters of mathematical model and the advantages of numerical methods, we present an efficient combination algorithm of the method of characteristics (MOC) and mixed finite volume element (MFVE). The MFVE is conservative, and could compute a function and its gradient at the same time. Thus, the MFVE is used for computing the pressure, Darcy velocity and the saturation. Furthermore, the saturation equation is convection-dominated. MOC may eliminate numerical dispersion and nonphysical oscillation, and computes the values at the sharp fronts well. Thus, an MFVE-MOC is formed for the saturation. Using the variation, energy estimates, induction hypothesis, embedding theorem and the technique of a priori estimates, we obtain an optimal error estimates in $l^2$ norm. Finally, numerical examples are given to show the effectiveness and practicability. The composite scheme possibly solves this actual problem well.

本文考虑了两相可压缩位移问题的数值模拟，以偏微分方程（PDE）的非线性系统为模型。考虑到数学模型的特点和数值方法的优点，提出了一种有效的特征方法（MOC）和混合有限体积元（MFVE）组合算法。MFVE是保守的，可以同时计算一个函数及其梯度。因此，MFVE用于计算压力，达西速度和饱和度。此外，饱和度方程是对流主导的。MOC可以消除数值离散和非物理振荡，并可以很好地计算出锋利前沿的值。因此，形成用于饱和的MFVE-MOC。使用变化，能量估计，归纳假设，${升}^2$规范。最后，通过数值算例表明了该算法的有效性和实用性。复合方案可能很好地解决了这个实际问题。