In this paper, we develop high-order bound-preserving (BP) finite difference (FD) methods for the coupled system of compressible miscible displacements. We consider the problem with multi-component fluid mixture and the (volumetric) concentration of the jth component, $c_j$, should be between 0 and 1. It is well known that $c_j$ does not satisfy a maximum-principle. Hence most of the existing BP techniques cannot be applied directly. The main idea in this paper is to construct the positivity-preserving techniques to all $c_j^{\prime }s$ and enforce ${\sum }_j{c}_j=1$ simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure $dp/dt$ as a source in the concentration equation and choose suitable “consistent” numerical fluxes in the pressure and concentration equations. Recently, the high-order BP discontinuous Galerkin (DG) methods for miscible displacements were introduced in [6]. However, the BP technique for DG methods is not straightforward extendable to high-order FD schemes. There are two main difficulties. Firstly, it is not easy to determine the time step size in the BP technique. In finite difference schemes, we need to choose suitable time step size first and then apply the flux limiter to the numerical fluxes. Subsequently, we can compute the source term in the concentration equation, leading to a new time step constraint that may not be satisfied by the time step size applied in the flux limiter. Therefore, it would be very difficult to determine how large the time step is. Secondly, the general treatment for the diffusion term, e.g. centered difference, in miscible displacements may require a stencil whose size is larger than that for the convection term. It would be better to construct a new spatial discretization for the diffusion term such that a smaller stencil can be used. In this paper, we will solve both problems. We first construct a special discretization of the convection term, which yields the desired approximations of the source. Then we can find out the time step size that suitable for the BP technique and apply the flux limiters. Moreover, we will also construct a special algorithm for the diffusion term whose stencil is the same as that used for the convection term. Numerical experiments will be given to demonstrate the high-order accuracy and good performance of the numerical technique.

在本文中，我们为可压缩混溶位移耦合系统开发了高阶保界（BP）有限差分（FD）方法。我们考虑了多组分流体混合物和第j个组分的（体积）浓度的问题，$C_{Ĵ}$，应介于0到1之间。众所周知， $C_{Ĵ}$不满足最大原则。因此，大多数现有的BP技术无法直接应用。本文的主要思想是为所有人构建阳性保留技术$C_{Ĵ}^{\prime }s$ 并执行 ${\sum }_{Ĵ}{C}_{Ĵ}=\mathrm{1个}$同时获得与物理相关的近似值。这样做，我们必须处理压力的时间导数$dp/dŤ$作为浓度方程式的来源，并在压力和浓度方程式中选择合适的“一致”数值通量。最近，在[6]中引入了用于混溶位移的高阶BP不连续Galerkin（DG）方法。但是，用于DG方法的BP技术不能直接扩展到高阶FD方案。主要有两个困难。首先，在BP技术中确定时间步长并不容易。在有限差分方案中，我们需要首先选择合适的时间步长，然后将通量限制器应用于数值通量。随后，我们可以在浓度方程中计算源项，从而导致新的时间步长约束，而通量限制器中应用的时间步长可能无法满足该约束。所以，确定时间步长将非常困难。其次，对于扩散项的一般处理，例如在可混溶位移中的中心差，可能需要一个模版，该模版的尺寸大于对流项的模版。最好为扩散项构造一个新的空间离散化，以便可以使用较小的模板。在本文中，我们将解决这两个问题。我们首先构造对流项的特殊离散化，从而得出源的所需近似值。然后，我们可以找到适合BP技术的时间步长，并应用磁通限制器。此外，我们还将为扩散项构造一个特殊的算法，该算法的模板与对流项的模板相同。