Comparison of Fixed Point Methods and Krylov Subspace Methods Solving Convection-Diffusion Equations

The paper first introduces two-dimensional convection-diffusion equation with boundary value condition, later uses the finite difference method to discretize the equation and analyzes positive definite, diagonally dominant and symmetric properties of the discretization matrix. Finally, the paper uses fixed point methods and Krylov subspace methods to solve the linear system and compare the convergence speed of these two methods.


Introduction
In the case of a linear system Ax b = , the two main classes of iterative methods are the stationary iterative methods (fixed point methods) [1]- [3], and the more general Krylov subspace methods [4]- [13].When these two classical iterative methods suffer from slow convergence for problems which arise from typical applications such as fluid dynamics or electronic device simulation, preconditioning [14]- [16] is a key ingredient for the success of the convergent process.
The goal of this paper is to find an efficient iterative method combined with preconditioning for the solution of the linear system Ax b = which is related to the following two-dimensional boundary value problem (BVP) [17]: , , , 0,1 0, , .
For parameter 0,16, 64 γ = , we use finite difference method to discretize the Equation (1).Take 2 n + points in both the x-and y-direction and number the related degrees of freedom first left-right and next bottomtop.Now, we take 32 n = , thus the grid size 1 1 2 1 33 . For 0 γ = , the convection term equals to 0, using central difference method to the diffusion term, we get the discretization matrix of the Equation (1) For 16 γ = , the diffusion term use a central difference and for the convection term use central differences scheme as the following: we obtain the discretization matrix of the Equation ( ) ) , the diffusion term use a central difference and for the convection term use upwind differences scheme as the following: we obtain the discretization matrix of the Equation (1) where ( )

Properties of the Discretization Matrix
In this section, we would first compute the eigenvalues of the discretization matrices 0 A , 1 A and 2 A of Equ- ation (1), later analyze positive definite, diagonally dominant and symmetric properties of these matrices.

Eigenvalues
Using MATLAB, the eigenvalues of the discretization matrices 0 A , 1 A and 2 A of Equation ( 1) are the fol- lowing:

Definite Positiveness
Matrix A is positive definite if and only if the symmetric part of A i.e.

A
(i = 0, 1, 2) is positive definite, which means the discretization matrices 0 A , 1 A and 2 A of Equation ( 1) are positive definite.

Diagonal Dominance
For all the discretization matrices 0 A , 1 A and 2 A of Equation ( 1), ( ) Therefore, the discretization matrices 0 A , 1 A and 2 A of Equation ( 1) are diagonally dominant.

Symmetriness
It is easy to see that only 0 A is symmetric.

Stationary Iteration Methods and Krylov Subspace Methods
The goal of this section is to find an efficient iterative method for the solution of the linear system , 0,1, are positive definite and diagonally dominant, we would use fixed point methods and Krylov subspace methods.In this section, first find the suitable convergence tolerance, later use numerical experiments to compare the convergence speed of various iteration methods.

Convergence Tolerance
Without loss of generality, take ( ) ( ) In order to achieve

Numerical Experiments
Computational results using fixed point methods such as Jacobi, Gauss-Seidel, SOR etc. and projection methods such as PCG, BICG, BICGSTAB, CGS, GMRES and QMR are listed out in figures (Figures 1-21), for all three different γ values ( ) 0,16, 64 γ = . The projection methods are performed with different preconditioning methods such as Jacobi preconditioning, luinc and cholinc preconditioning.
The tables (Table 1 and Table 2) containing relevant computational details are also given below.

Conclusions
From the figures (Figures 1-21) and tables (Table 1 and Table 2), we obtain the following conclusions: • The convergence speeds of SOR, Backward SOR and SSOR are faster than that of GS and Backward GS; while GS and Backward GS are faster than Jacobi; • The upwind difference method is more suitable to be applied to convection dominant problem than the cen-               tral difference method; • The convergence speed of the six projection methods including PCG, BICG, BICGSTAB, CGS, GMRES and QMR under luinc preconditioning are faster than under cholinc preconditioning, while under cholinc preconditioning are faster than Jacobi preconditioning; • The six projection methods under Jacobi, luinc and cholinc are convergent when 0 γ = , however, for 16, 64 γ = , the PCG method are not convergent and also the CGS method under Jacobi preconditioning are not convergent when 64 γ = .

,
the convergence speeds of SOR, Backward SOR and SSOR are the same; SSOR is faster than Backward SOR and SOR; • The convergence speeds of SOR and Backward SOR are the same, also for GS and Backward GS; • From Table2, the iteration steps and the time for all fixed point methods of the case in which 64 γ = are less than the case in which 16 γ = , and also the case in which 16 γ = are less than the case in which 0 γ = ;

Figure 1 .
Figure 1.Comparison of convergence speed using fixed point methods when = 0 γ .

Figure 2 .
Figure 2. Comparison of convergence speed using fixed point methods when = 16 γ .

Figure 3 .
Figure 3.Comparison of convergence speed using fixed point methods when = 64 γ .

Table 1 .
Computational details of different projection methods with different preconditioning.

Table 2 .
Computational details of different fixed point methods.