Parallel Iterative Algorithms with Accelerate Convergence for Solving Implicit Difference Equations

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INTRODUCTION
Recently, parallel algorithms with such good properties such unconditional stable schemes and higher accuracy schemes for solving implicit difference equations have been improved greatly. Both ASE-I algorithm and ASC-N algorithms, known as segment implicit methods, are set up to solve different implicit equations [9] . They realize the principle of divide and rule and efficient computation in parallel by segmenting grid domains. It turns out that iterative methods are of convergent properties, which was proved by splitting coefficient matrix [1] . Segment Classic Implicit Iterative (SCII) and Segment Crank-Nicolson Iterative (SC-NI) algorithms for solving one-dimensional diffusion equation, which can be decomposed into smaller tridiagonal subsystems, are solved by using a doublesweep algorithm [8] . The convergent rate is estimated and the property of gradual-approach convergence about one-dimensional implicit difference equations is discussed. The algorithms can solve implicit difference equations and have an efficiency in parallel [2,5] . For improving the convergence rates and the properties of gradual-approach convergence of the SCII and SCNI algorithms, SCII and SCNI algorithms with accelerated convergence which can be decomposed into smaller strictly tri-diagonally dominant subsystems and be solved by using a double-sweep algorithm are studied and improved through the inserting classic implicit scheme and Crank-Nicolson scheme into them respectively. General structures of SCII and SCNI algorithms with accelerated convergence are described by using matrix form. The improved convergent rates and properties of the gradual-approach are described by a splitting coefficient matrix in detail. By using the algorithms, it can save much time to solve implicit difference equations in parallel. In addition, the algorithms are extended to two-space dimensional problems by studying Peaceman-Rachford scheme into which classic implicit schemes are inserted alternately.
Finally, theoretical analysis and numerical exemplifications show that the parallel iterative algorithms with accelerated convergence are of higher efficiency in computation, have much better convergent rate and property of gradual-approach convergence.

SCII algorithm with accelerated convergence:
The problem is to find the solution of u(x,t) in the with the boundary conditions: and the initial condition as: Let ∆x and ∆t be the step sizes in the directions of x and t, where, 1 m x ∆ = , m is a positive integer. The approximate values u k i of the solution u(x,t) for the problems (1)-(3) are to be computed at the grid points (x i , t k ), where x i = i∆x, for I = 0,1, …, t k = k∆t for k =1,2...., For simplicity, we denote points (x i , tk) by (I, k).
In order to improve the convergence rate and property of gradual approach convergence, the classic implicit difference equation can be made as follows: The schemes above are embedded in difference Eq. 4-6 regular. So the numerical solutions involving finite difference representation of the equations governing diffusion processes usually consist of solving (m-1)×(m-1) system which may be written in the matrix form as: k 1 k 1 k 1 k 1 T 1 2 m 1 (u ,u ,...,u ) where, b is implicated by U k Generally, we resort to solve (7) which is based on the splitting of the matrix A as follows: where, M and Nare given respectively by:  -r -r 1 2r -r -r 1 r -r -r 1 2r  [k] in computing in this study. By splitting off (8), a SCII algorithm with accelerated convergence can be expressed as: To balance the computing in parallel, m i for i = 1, 2,…, k is often made equal in performing. Since A i for i = 1, 2,…,k has been strictly tri-diagonally dominant and N i for i = 1,2,….,k-1 has only one non-zero element, massive computing in parallel about (9) is not difficult by parallel segmented double-sweep algorithm [7,8] .
Analyses of convergent rate and property of gradual-approach convergence: In this section, stability, convergent rate and property of gradual approach convergence of (9) are analyzed. We firstly introduce a famous lemma as follows [3,4] .
It is easy to prove that (7) is unconditionally stable according to the above lemma. And the estimate result in the iteration of (9) is what follows: (9) is convergent. Obviously, the convergence rate of (9) is better than that of SCII algorithm [2] .
The property of gradual approach convergence will be discussed in detail in the following. For balancing the computing in parallel, we suppose A i = A p for i= 1,2,…,k. So we have: The result of -1 p p p ¥ A (M + N ) will be estimated in the following section. (11) can be obtained by solving the first and last linear system of: By Grammer's rule, it is easy to get the two vectors X 1 and X p as: Taking i as continuous and differentiable in f(I,p), we have: In the following section, Namely: Let f(I,p)=y i,p and Y j = (y 1 ,j,y 2 ,j,…,y pj ) T for i = 1,2,..,p. We have: In conclusion, we have:  [5,10] . Therefore, we have: Theorem: Segment classic implicit iterative algorithm with accelerated convergence (9) for solving diffusion Eq. 1-3 is convergent. Its convergent rate is better than SCII algorithm, and the property of gradual approach convergence can approach 2 r ( ) α gradually with the increasing of net point number in each segment.

SCNI algorithm with accelerated convergence:
It is well known that Crank-Nicolson scheme of (1) has unconditional stability and has much better accuracy as follows: k 1 k 1 k 1 k k k j 1 j j 1 j 1 j j 1 ru 2(r 1)u ru ru 2(1 r)u ru Similar to the transfiguration of the classic implicit scheme, Crank-Nicolson scheme can be changed in the following equation: Equation (17) In which, the structure of N i and M i are similar to (8) ≤ . In addition, its convergent rate, property of gradual approach convergence and accuracy are much better than those of SCII algorithm with accelerated convergence.

A Parallel iterative algorithm with accelerated convergence for two-dimensional problems:
The problem is to find the solution u (x, y, t) in the domain with the boundary conditions: and the initial condition: u(x,y,0)=f(x,y) sometimes denote points (I, j, k) by (x i , y j , t k ). Among the finite difference method for the numerical solution above, the well-known Peaceman-Rachford scheme is unconditionally stable and has truncation error 2 2 ( t x ) Ο ∆ + ∆ as follows: ru (1 2r)u ru u r(u 2u u ) with the boundary conditions: for I, j =0,1…m and the initial condition: where, 2 / h r τ = . So the difference Eq, 24-26 for the Eq. 21-23 consist of solving (m-1)×(m-1) system, which can be written in the matrix form as: where, A is defined as: 1 2r r 0 r 1 2r r r 1 2r r 0 r 1 2r It is well known that Peaceman-Rachford schemes can be divided into two processes in alternate directions. So parallel iterative algorithm with accelerated convergence of (21)-(23) can be expressed as: where, M and N are defined as (11). By a similar method in one-dimensional problems, the convergence rate and property of gradual approach convergence are estimated as follows: It is not difficult to see that parallel iterative algorithm with accelerate convergence (28) for twodimensional problems are convergent. In addition, it has a better convergent rate and property of gradual convergence than those of parallel iterative algorithm about two-dimensional diffusion problem [5] . Thus, the parallel iterative method with accelerated convergence for one-dimensional problem is extended to the computation of multi-dimensional problem.

Numerical examples and numerical results:
One-dimension example: Consequently, numerical experiments are made for problems of (1)- (3) in which: The exact solution of the problem is:         The numerical results are shown in Table 1  The numerical results of SCII and SC-NI algorithms with accelerated convergence obtained in SGL/Challenge L with 8 CPUs for these methods are more accurate in computing in parallel. It is shown in Table 2 that the iterative degree of SCII algorithm with accelerated convergence is less than that of SCII algorithm and it decreases with the increasing of net point number in each segment (Table 3).

≤ ≤
The exact solution of the problem is:

CONCLUSION
By reconstructing differential equations, the SCII algorithm with accelerated convergence for solving one-dimensional diffusion equation is developed in this study. It is convergent in iteration and has a better convergent rate and property of gradual approach convergence than those of SCII algorithm. Furthermore, the SC-NI algorithm with accelerated convergence is discussed. The convergent rate, property of gradualapproach convergence and accuracy are much better than those of SCII algorithm with accelerated convergence. In addition, the algorithm is tended to two-space dimensional problem by studying Peaceman-Rachford schemes.