Barycentric Rational Collocation Method for the Incompressible Forchheimer Flow in Porous Media

Barycentric rational collocation method is introduced to solve the Forchheimer law modeling incompressible fluids in porous media. -e unknown velocity and pressure are approximated by the barycentric rational function. -e main advantages of this method are high precision and efficiency. At the same time, the algorithm and program can be expanded to other problems. -e numerical stability can be guaranteed. -e matrix form of the collocation method is obtained from the discrete numerical schemes. Numerical analysis and error estimates for velocity and pressure are established. Numerical experiments are carried out to validate the convergence rates and show the efficiency.


Introduction
Darcy flow in porous media is of great interest in many science and engineering fields such as oil recovery and groundwater pollution contamination. Darcy's law, mainly describes the linear relationship between Darcy velocity u and derivative of pressure p. Here, symbols μ, k, ρ, and g(x) represent the viscosity coefficient, permeability, the density of the fluids, and the gravitational term, respectively. is model is widely used and suitable for low velocity, small porosity, and permeability fluids [1][2][3][4].
If the porosity is nonuniform and velocity is higher, a second-order term is needed to be added, the non-Darcy relationship has been researched by Forchheimer [1]. For example, the high-speed Forchheimer flow of single-phase incompressible fluid in porous medium is presented as follows: Note that when Forchheimer number β � 0, nonlinear model (2) degenerates to linear Darcy's law (1).
Model (2) is also called Darcy-Forchheimer law [5][6][7][8][9][10]. In [6], a block-centered finite difference method has been introduced to solve the Darcy-Forchheimer law. Discrete numerical scheme and error estimates were given. Mixed finite element method (MFEM) for equation (2) was studied in [7,8]. Using this method, velocity and pressure can be approximated simultaneously. Two-grid and multigrid block-centered finite difference method (FDM) for the Darcy-Forchheimer flow in porous media was researched in [10,11], respectively. is method can improve the efficiency of dealing with nonlinear problems. e barycentric formula is obtained by the Lagrange interpolation formula [12][13][14][15][16] and has been used to solve Volterra equation and Volterra integro-differential equation [12,17,18]. Floater and Hormann [19] have proposed a rational interpolation scheme which has higher accuracy on equidistant and special distributed nodes. Wang et al. [20][21][22] successfully applied the barycentric rational collocation method (BRCM) to solve initial value problem, boundary value problem, plane elasticity problem, and some nonlinear problems. ese research studies extended the application fields of barycentric rational collocation method. In recent papers, Li et al. [23][24][25][26][27] have used the barycentric rational collocation method to solve heat conduction equation, biharmonic problem, and second-order Volterra integro-differential equation.
In this paper, barycentric rational collocation method is introduced to solve the incompressible Forchheimer flow. We demonstrate that barycentric rational collocation method is highly accurate for both velocity and pressure. O(h d ) error estimates for velocity and pressure are given. Numerical experiments [28][29][30][31][32] are carried out to show the convergence rates. e paper is organized as follows. In Section 2, notations and barycentric formula are given. In Section 3, convergence analysis of barycentric rational collocation method for Forchheimer law and error estimates of velocity and pressure are presented. In Section 4, numerical examples are carried out to verify the convergence rates and show the efficiency. roughout this paper, C denotes a positive constant independent of h.

Notations and Barycentric
Rational Algorithm e partition of interval Ω � [a, b] is as follows: Define For the function u(x), the interpolation function r(x) (d � 0, 1, . . . , n) is given as Symbol p i (x) denotes the d-order interpolation polynomial such that p i (x k ) � u(x k ) for k � i, i + 1, . . . , i + d, where u k � u(x k ) and λ i (x) is a blending function For the numerator term in (5), we deduce that Here, and and for the denominator term in (5), rough further deduction, we get Here, ω k is described as (9). e basis function r j (x) of barycentric rational interpolation is en, we get the derivative formula at node x i as Its matrix formulation can be given as where e derivative formulation of the basis function r j (x) at node x i is 2 Journal of Mathematics According to induction (14)- (18), we obtain the recurrence formula of D (m) ij as

Convergence Rates and Error Estimates
Define the error between u(x) and barycentric rational interpolation function r(x) as According to rational interpolation error theory, we know Combing (21) with (5), we see where (23) Define the error norm of e(x)as e following lemma has been proved in [12].
Lemma 1 (see [12]). For the error e(x) defined as (20), we have Now, we deal with the barycentric rational collocation schemes for the following Forchheimer equations: For the second equation of (26), the approximate formula is (27), the numerical scheme is For the first equation of (26), the approximate formula is as follows: en, the calculation scheme is Note that, in practical calculation, first step, we approximate the second equation of (26) and then the first equation of (26).
Let u(x n ) denote the numerical solution of u(x), then we have Based on the above states, the next theorem gives the error analysis of Darcy velocity.
Proof. For the second equation of (26), using the notation of differential matrix, the discrete form of the collocation method is where Journal of Mathematics 3 (34) Furthermore, we have e proof of this theorem is completed. Let p(x n ) denote the numerical solution of p(x), then we have e following theorem presents the error analysis of pressure p.
Proof. For the first equation of (26), the discrete numerical scheme is where Furthermore, we see As E 1 , note that μ and k are positive constants, we have Similarly, for E 2 , according to the monotonicity of the nonlinear term, we know Combing (40)-(42), the proof is finished.

Remark 1.
In the above proof of eorem 2, coefficients μ, k, β, and ρ are supposed to be positive constants. If they are functions that depend on variable x and bounded, the proof is similar.

Numerical Experiments
In this section, we carry out some numerical experiments using barycentric rational collocation method to solve the Forchheimer equations.
e analysis solution is chosen to be u(x) � 2 sin(πx), Gravitational term g(x) is determined according to the first equation of (43). Define absolute error and relative error as Numerical results are listed in Tables 1-4 . e corresponding approximate figures between analysis solution and numerical solution can be seen in Figures 1 and 2 e analytical solution is set to be p(x) � (2x + cos(7πx))ln(x + 1).

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.