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A new compact high order off-step discretization for the system of 2D quasi-linear elliptic partial differential equations
Advances in Difference Equations volume 2013, Article number: 223 (2013)
A new fourth-order difference method for solving the system of two-dimensional quasi-linear elliptic equations is proposed. The difference scheme referred to as off-step discretization is applicable directly to the singular problems and problems in polar coordinates. Also, new fourth-order methods for obtaining the first-order normal derivatives of the solution are developed. The convergence analysis of the proposed method is discussed in details. The methods are applied to many physical problems to illustrate their accuracy and efficiency.
We consider the two-dimensional (2D) quasi-linear elliptic partial differential equation (PDE) of the type
where , with boundary ∂R (see Figure 1), subject to the Dirichlet boundary conditions given by
The PDEs of the type (1) with variable coefficients model many problems of physical significance. For instance, the convection-diffusion and Burgers’ equations that represent the transport phenomena, and the highly nonlinear Navier-Stokes’ (N-S) equations of motion that describe the motion of fluid flow and represent the conservation of mass, momentum and energy.
We make the following assumptions about the boundary value problem (1):
f is continuous,
where H and I are positive constants and is the set of all functions of x and y with continuous partial derivatives up to order m in the region R. The condition (a) guarantees the ellipticity of equation (1). Conditions (e), (f) and (g) are the necessary conditions for the existence and uniqueness of the solution of boundary value problem (1)-(2) (see ).
A number of high order compact schemes have been reported for the linear elliptic problems like Poisson’s equation and the convection diffusion equation (see [2–9]). Ananthakrishnaiah and Saldanha  framed a 13-point fourth-order compact scheme for the solution of a scalar nonlinear elliptic PDE, which was later extended to a system of equations by Saldanha . The finite difference methods for solving the steady state incompressible N-S equations vary considerably in terms of accuracy and efficiency. It has been discovered that although central difference approximations are locally second-order accurate, they often suffer from computational instability and the resulting solutions exhibit non-physical oscillations. The upwind difference approximations, though computationally stable, are only first-order accurate and the resulting solutions exhibit the effects of artificial viscosity. A number of high order compact schemes for the solution of the N-S equations in stream function vorticity form in the Cartesian coordinates were proposed in [12–15]. In 1997, Mohanty  proposed fourth-order difference methods for 2D nonlinear elliptic boundary value problems with variable coefficients using only nine grid points of a single computational cell. This method could be successfully applied to the N-S model equations in polar coordinates. Later, Mohanty and Dey  developed the fourth-order accurate estimates of the first-order normal derivatives of the solution viz . However, the methods  and  could not be directly applied to singular elliptic problems, and they required suitable modifications at the points of singularity. In this regard, Mohanty and Singh  derived an off-step fourth-order discretization for the solution of singularly perturbed two-dimensional nonlinear elliptic problems and the estimates of , which were directly applicable to singular elliptic problems.
In this article, we develop new off-step fourth-order discretizations for the solution of the system of quasi-linear elliptic PDEs with variable coefficients, and the estimates of , using the nine grid points of a single computational cell (see Figure 1). The main advantage of the proposed methods is that they are directly applicable to the singular problems and the problems in polar coordinates, without any need of modifications, hence reducing the manual and mechanical calculations reasonably.
An outline of the article is as follows: In Section 2, we discuss and derive the off-step fourth-order compact discretization schemes for the solution of a nonlinear elliptic equation with variable coefficients and the estimates of . These methods are further extended to the solution of the quasi-linear PDE given by (1)-(2). In Section 3, we establish the fourth-order convergence of the method for a scalar equation under appropriate conditions. Further, in Section 4, the stability analysis of the steady state convection diffusion equation is conducted. In Section 5, we generalize our methods for the system of quasi-linear PDEs with variable coefficients, subject to the Dirichlet boundary conditions. In Section 6, we implement the proposed methods over linear and nonlinear problems of physical significance to illustrate and examine the accuracy of these methods. Section 7 contains some concluding remarks about this article.
2 The off-step discretization and derivation
We first consider the following two-dimensional nonlinear elliptic PDE:
for , subject to the Dirichlet boundary conditions given by (2).
We superimpose on the domain R a rectangular grid with spacing in both x and y-directions. Let us introduce the following notations:
Each grid point is given by or simply for and , , where .
Further, at each grid point , let:
and denote the exact and approximate values of , respectively.
For , let
Then, for , differential equation (3) can be written as
For the fourth-order discretization of PDE (3), we simply follow the approach given by Chawla and Shivakumar .
We set the following approximations:
where s, s and s () are the parameters to be suitably determined.
Then, at each internal grid point , differential equation (3) is discretized by
for , where we denote
for and being the central and average difference operators in x-direction etc.
Now, with the help of Taylor series expansion, it is easy to obtain
Using (6.1) and (6.2) and simplifying (5.1)-(5.8) by Taylor series expansions, we obtain
Using equations (11.1), (11.2), simplifying (5.9)-(5.12) and (7.1)-(7.3), we obtain
Finally, from (8), using (12.1)-(12.3), we obtain
Substituting approximations (11.1), (11.2) and (13) into (9), and by the help of (10), we obtain
Thus, for the proposed difference method (9) to be of fourth order, the coefficient of in (14) must be zero, and hence we have
Equating to zero the coefficients of each of , and , we obtain the values of the unknown parameters as follows:
thereby reducing to . Thus the difference method of for nonlinear PDE (3) is given by (9) for the above values of parameters.
Now, we consider the numerical method of for the solution of 2D quasi-linear elliptic equation (1). In order to understand the concept to develop the method for the quasi-linear case, we consider the following differential equation:
A fourth-order method for differential equation (15) is given by
where , and .
Whenever the differential equation (15) is of the form , the evaluation of is difficult and formula (16) needs to be modified. Substituting in (16), we obtain the modified version of (16) due to Numerov as
where . Note that (17) is consistent with the differential equation .
Now, we use the above concept to derive the numerical method for quasi-linear equation (1). Since the coefficients are the functions of not only the independent variables x and y but also of the dependent variable u, i.e., and , the difference scheme (9) cannot be applied directly as the first- and second-order derivatives of u are unknown at the internal grid points. Thus further discretizations of , , and are required in the method (9) without affecting its order. For this purpose, for , we use the following central differences:
Upon substitution of the central differences (18.1)-(18.4) in the method (9), it is easy to verify that
We observe that the truncation error retains its order , and hence we obtain the required numerical method of for the solution of quasi-linear elliptic PDE (1).
After having determined the fourth-order approximations to the solution of equation (3), we now discuss the fourth-order numerical methods for the estimates of and . One may compute these values using the standard central differences:
It is found that the standard central differences (19.1) and (19.2) yield second-order accurate results irrespective of whether fourth-order difference method (9) or standard difference scheme is used to solve PDE (3). Thus, new difference methods for computing the numerical values of and are developed, which are found to yield accurate results when used in conjunction with the nine-point formula (9).
At each grid point , we denote the exact and the approximate solutions of , by , and , , respectively. Then, following the techniques given by Stephenson , for , we obtain
A simple Taylor series expansion would yield
Then, using equations (11.1) and (21) in (20.1), we obtain . Similarly, we obtain . Hence, equations (20.1)-(20.2) give the fourth-order approximation to the first-order normal derivatives of the solution of nonlinear equation (3). The numerical methods (20.1)-(20.2) are applicable when the fourth-order numerical solutions of u are known at each internal grid point. Further, the Dirichlet boundary conditions are given by (2). The difference method (9) for the determination of u can be easily expressed in tri-block-diagonal matrix form, and the methods (20.1)-(20.2) for determination of and can be expressed in diagonal matrices form, thus can be easily solved. The proposed methods (9), (20.1) and (20.2) are directly applicable to singular elliptic problems in the region R.
Now, for the two-dimensional quasi-linear elliptic equation (1), using the approximations (18.1) and (18.2) in (20.1)-(20.2), we easily obtain
where and are of .
3 Convergence analysis
We consider the 2D nonlinear elliptic partial differential equation
defined in the region R, subject to , , where are constants.
Then the difference method (9) for equation (23) is given by
Let, for each such that ,
Also, for , let
where t denotes the transpose of the matrix.
Then, varying such that , equation (24) may be written in the matrix form as
We assume here that and . Thus, all the diagonal entries of matrix D are positive and all the off-diagonal entries are negative.
Since U is the exact solution vector, we have
where for each such that .
We may write
for suitable , and , where .
Also, for , we may write
With the help of equations (27.1)-(27.3) and (28.1)-(28.4), we obtain
where [, ] is the tri-block diagonal matrix with
and for , let
for some positive constants Q, and .
Now, it is easy to verify that for sufficiently small h,
Further, the directed graph of shows that it is an irreducible matrix (see Figure 2). The arrows indicate the paths for every nonzero entry of the matrix . For any ordered pair of nodes i and j, there exists a direct path connecting i to j. Hence, the graph is strongly connected. So, the matrix is irreducible (see Varga ).
Let denote the sum of the elements in the k th row of , then for , we have
And finally, for , ,
With the help of equations (31.1)-(31.5), we get
It follows that for sufficiently small h,
Thus, for sufficiently small h, is monotone. Hence exists and (see Henrici ), where
and , from equations (32.1)-(32.4), with , it follows that
Equation (30) may be written as
This establishes the convergence of the fourth-order difference method (9) (with ) for the scalar elliptic equation (23).
4 Stability analysis
We consider the steady state two-dimensional convection-diffusion equation
where is a constant, with ε (the perturbation parameter) being the ratio of convective velocity to the diffusion coefficient.
Applying the difference scheme (9) with to the above equation and letting , which is called the cell Reynolds number, we obtain
The above is a system of number of linear equations in number of unknowns, which may be expressed in the matrix form as , where
Now, applying the Jacobi iteration method to the above system of equations, we obtain
We examine the stability of (39) by assuming that an error exists at each grid point at the s th iteration. We analyze the behavior of the error by assuming it to be of the form
where A and B are arbitrary constants and ξ is the propagating factor which determines the rate of growth or decay of the errors. The necessary and sufficient condition for the iterative method to be stable is .
Using (40) in (39), the propagating factor for the Jacobi iteration method is obtained as
Thus, the Jacobi Iteration method is stable for those values of τ such that .
Similarly, applying the Gauss-Siedal iteration method to (38) and assuming the error at each grid point at the s th iteration to be of the form (40), the corresponding propagation factor is given by the equation
where , and .
Thus, the Gauss-Siedal iteration method is stable for those values of τ such that .
5 Generalisation of the above methods
We now extend our methods to the system of 2D quasi-linear elliptic PDEs of the form:
for , with each , and , subject to the Dirichlet boundary conditions given by
We assume and to be the exact and approximate values of respectively. For each , letting , we set the following approximations:
and finally, we define
Then, at each internal grid point , the fourth-order off-step discretization to each differential equation of system (43) is given by
for , where we denote
After the fourth-order approximate solution to system (43) is determined upon solving the tri-block diagonal system of equations (49), it is easy to see that the fourth-order estimates of can be explicitly obtained using the following discretizations:
where and are of .
6 Computational implementation
We implement the proposed method over three linear and seven nonlinear problems, including a quasi-linear problem, in Cartesian and polar coordinates. The exact solutions of the problems are given. The right-hand side functions and the Dirichlet boundary conditions are determined using the exact solutions. The system of linear difference equations is solved using the block iterative method and the system of nonlinear difference equations by the Newton-Raphson method (see Hageman and Young , Kelly  and Saad ). The iterations are terminated once the absolute error tolerance ≤10−12 has been reached. All the computations are done using MATLAB programming language.
Example 1 (Convection-diffusion equation)
subject to the Dirichlet boundary conditions given by
Example 2 (Poisson’s equation in r-θ plane)
At , the above equation represents 2D Poisson’s equation in cylindrical and spherical coordinates, respectively. The exact solution is .
Example 3 (Poisson’s equation in r-z plane)
At , the above represents the two-dimensional Poisson’s equation in cylindrical polar coordinates in r-z plane. The exact solution is .
Example 4 (Burger’s equation)
The exact solution is . The MAE in u, and are listed in Table 4 for .