Numerical Solutions of Two-Dimensional Vorticity Transport Equation Using Crank-Nicolson Method

: This paper is concerned with the numerical solutions of the vorticity transport equation (VTE) in two-dimensional space with homogenous Dirichlet boundary conditions. Namely, for this problem, the Crank-Nicolson finite difference equation is derived. In addition, the consistency and stability of the Crank-Nicolson method are studied. Moreover, a numerical experiment is considered to study the convergence of the Crank-Nicolson scheme and to visualize the discrete graphs for the vorticity and stream functions. The analytical result shows that the proposed scheme is consistent, whereas the numerical results show that the solutions are stable with small space-steps and at any time levels.


Introduction:
This work is concerned with the twodimensional vorticity-transport-equation (VTE), which is a nonlinear time-dependent partial differential equation: for ( , ) ∈ , > 0, with the following initial and boundary conditions:  (2). Due to the various applications of time-dependent partial differential equations in various fields of science, since last century, many authors have been interested in studying the analytical and numerical solutions of such types of problems including linear equations, nonlinear equations, partial integrodifferential equations, and time-space fractional-order partial differential equations, see for instance [1][2][3][4][5] . In fluid dynamics, the numerical solutions of various Mathematical models, including problem (1)- (2), have been studied by some authors, see for instance 6,7 .
It is known that problem (1)-(2) is used to study the unsteady flow problem in twodimensional space. In other words, it can be used for solving the two-dimensional viscous incompressible flow. In addition, the twodimensional vorticity transport equation can be used in some applications, such as analysis of laminar to turbulent flow transition, studies on free and mixed convection and the modeling of turbulent flows. For more details about the importance, derivation and the applications of this problem, see 8,9 . In fact, this problem cannot be solved analytically due to the nonlinear terms that appear in equation (1). So that since the last decades, problem (1)-(2), with different initial-boundary conditions, has been solved numerically by some authors using several methods, such as the Petrov-Galerkin finite element method 10 , finite difference schemes, see for instance [11][12][13][14][15][16] , and the boundarydomain integral method 17 . Because of the poor stability properties of explicit finite difference methods, the implicit methods are more recommended to compute the numerical solutions of initial-boundary value problems in two or more dimensions-space. The Crank-Nicolson method is one of the most recommended implicit methods for solving many types of second order linear problems with constant coefficients due to its high order of convergence and unconditional stability. However, it is not always guaranteed that Crank-Nicolson method is stable and applicable for other types of problems such as nonlinear problems, problems with variable coefficients and problems with nonlinear boundary conditions. In this work, the Crank-Nicolson finite difference scheme is used to solve problem (1)-(4). Moreover, it is shown that the proposed scheme is consistent and stable.
This paper is divided into seven sections. In the second section, the discrete formulas of equations (1) and (2), using Crank-Nicolson scheme, are derived. In the third section, the matrix forms of the Crank-Nicolson finite difference equations are presented. The consistency of the discrete difference equations is studied in the fourth section. In the fifth section, the stability condition for the matrices form is discussed. In the sixth section, the Crank-Nicolson discrete scheme is used to compute the numerical solutions of problem (1)-(4) with a certain initial function and a fixed value to the Reynolds number. Moreover, the numerical simulations for the vorticity and stream functions are shown in two-dimensional spaces and at different time levels. Finally, some conclusions and future works are stated in the seventh section.

Stability of the Discrete Problem
In this section, the stability for the matrix form (11) and (12) are discussed. The matrix form Crank-Nicolson scheme (11) and (12) can be rewritten as follows:

Numerical Experiment
The Crank-Nicolson difference equations (6) and (7) are used in this section to find the numerical solution of problem (1)-(4), with = 1, and the following initial function: (19) Moreover, in order to study the numerical convergence, different space-steps (ℎ = 0.4,0.2, 0.1) and a small fixed time-step = 0.002 are considered in the computations. Based on the type of the initial function (19), the solution of problem (1)-(4) with (19) is symmetric and positive. Therefore, it is sufficient to find only In addition, for each of ℎ = 0.4, ℎ = 0.2, and at the time level , the errors bounds will be computed that show, at some fixed meshes-points, the differences between the numerical solutions ( ℎ , ℎ ) and ( ℎ/2 , ℎ/2 ) with respect to ℎ and ℎ/2, respectively, as follows:
From Tables 1-3, it is observed that the numerical values for vorticity and stream are decreasing as time level increases. In addition, Table 4 shows that at a fixed time level, the corresponding error bounds decrease, as the space grids are refined. This indicates that the numerical solution is convergent. On the other hand, at any fixed space-step, the corresponding errors decrease as time increases. Moreover, Table 5 shows that the numerical results are stable (condition (17) is satisfied) with any space-step and time level.

Numerical Simulations
The discrete graphs of vorticity and stream functions (for ℎ = 0.1) at time levels = 0, 200 and 400 are presented in Figures 1, 2 and 3, respectively. Clearly, by Figs. 1-3, it is observed that the discrete graphs for vorticity and stream are decreasing as time increases and that supports the numerical results.

Conclusions:
This paper is concerned with the numerical solutions of the vorticity transport equation with homogenous Dirichlet boundary conditions using Crank-Nicolson finite difference scheme. From this work, the following conclusions are pointed out: 1-Crank-Nicolson finite difference scheme is consistent. Moreover, the order of the local truncation error has the form: ( 2 + ℎ 2 ). 2-At a fixed time level, the corresponding error bounds decrease, as the space grids are refined. This indicates that the numerical solution is convergent. 3-At any fixed space-step, the corresponding errors decrease as time increases. 4- Table 5 shows that the numerical results are stable with any space-step and time level. 5-Tables (1-3) and Figures (1-3) show that the numerical values for vorticity and stream are decreasing as time level increases. For future work, the following directions may be considered: 1. Other finite difference schemes can be proposed to find the numerical solution of problem (1)-(4), such as implicit Euler scheme. 2. One could solve problem (1)-(4), with a certain initial function using different consistent finite difference schemes including the present one in order to make a numerical comparison between the results regarding stability and error bounds. 3. With a very large Reynolds number, the nonlinear terms in equation (1) are dominated, so that may affect the stability properties of the proposed scheme. Therefore, in this case, other numerical methods should be adapted to solve the problem.