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The compressible miscible displacement in a porous media is considered in this paper. The problem is a nonlinear system with dispersion in non-periodic space. The concentration is treated by a characteristics collocation method, and the pressure is treated by an orthogonal collocation method. Optimal order estimates are derived.

The mathematical controlling model for compressible miscible displacement in porous media with dispersion is given by

where

We shall assume that no flow occurs across the boundary

where v is the outer normal to

The compressible flow problems are strongly nonlinear coupling system for partial differential equations of two different types, and we consider the system with dispersion in non-periodic space, so these factors lead to many difficulties for convergence analysis of algorithms. The collocation methods are widely used for solving practice problems in engineering due to its easiness of implementation and high-order accuracy. But the most parts of mathematical theory focused on one-dimensional or two-dimensional constant coefficient problems [

The organization of the rest of the paper is as follows. In Section 2, we will present the formulation of the characteristic collocation scheme for nonlinear system (1) (2). In Section 3, we will analyze convergent rate of the scheme defined in Section 2.

In this subsection, we will give some basic notations and definition for the characteristics collocation methods, which will be used in this article. We make the partition of the domain

Define function spaces as follows:

where

Next, we take four Gauss points as collocation points in

Let

In this subsection we will present the fully discrete characteristic collocation scheme for nonlinear system (1) (2) with dispersion term in non-periodic space. At first time

The Equation (2) can be put in the form

For (5), we use a backward difference quotient for

where

So we can obtain the following discrete equation:

Now that use the interpolation operator

and

for

We can understand the following method intuitively from above schematic diagram. When

In this section we consider the existence and uniqueness of the numerical solution, and obtain the optimal error estimate. CCS (8) (9) can be rewritten as the discrete Galerkin method given by [^{ }

We can get the following convergence conclusion for the above numerical Scheme (11) (12).

Theorem 3.1 Suppose

Proof: Let

where

We start this induction by seeing that

for

Next we will consider the concentration equation, subtracting (12) from the discrete Galerkin scheme of (2),

To obtain optical estimate for

If

for h sufficiently small and

At last we shall check the induction hypotheses (14) and (17)

for h sufficiently small , and the proof is complete.

We thank the fund “Basic Subjects Fund of China University of Petroleum (Beijing) (KYJJ2012-06-04)”.

Ning Ma,Xiaofei Lu, (2015) Characteristics Collocation Method of Compressible Miscible Displacement with Dispersion. Journal of Applied Mathematics and Physics,03,86-91. doi: 10.4236/jamp.2015.31012