INTRODUCTION
In recent years, with the need of great scale scientific computing and the
maturity of parallel computing environments, domain decomposition method (Quarteroni
and Valli, 1999; Dawson and Dupont, 1992; Marini
Quarteroni, 1988; Mathew et al., 1998) has
been an effective approach to solve partial differential equations numerically.
There are a lot of good work in this field: Dawson et
al. (1991), Wan et al. (2002), who made
domain decomposition algorithm of the explicitimplicit scheme to let out D^{2}
times over than the classical explicit scheme on the restriction of the stability
and the convergence that is ,
but the accuracy achieved .
The idea in Zhang (1998) is almost same with Dawson
et al. (1991), however, the former used explicit scheme of Rodrigue
and Wolitzer on the interface point. The stability condition is Δt/h^{2}≤2;
the error is o(Δt+h^{2}). Peaceman and Rachford
(1995) and Zhou and Yuan (1999) proposed the parallel
difference format for parabolic equation with the error .
The reference Zhang and Shen (2002) used the Saul’yev
asymmetric difference scheme which is adopt the stride step H to use in the
interface point, furthermore, constructed a new domain decomposition algorithm
and give a prior error estimate which
is in circumstances of two subregions and the stable condition is .
In order to improve the accuracy of the interface point, the reference Du
et al. (2001) considered the multistep explicit scheme and the onestep
highlevel explicit scheme.
Domain decomposition algorithm for onedimensional heat equation: Zuo
and Yuan (2003) proposed a new domain decomposition algorithm for onedimensional
heat equation by using Du FortFrankel scheme at the interface points and the
fully implicit scheme at the interior points.
Firstly problem of solving the onedimensional heat equation is considered:
where, u°(x) is a given function.
For simplicity, at first we divide domain (0, 1) into only two subdomains
(0, )
and (,
1). Suppose N, M, D are positive integers. Let h=1/N and x_{i} = i_{h},
i = 0, ..., N. Take τ = Δt = T/M and t^{n} = nΔt, with
the assumption that = x_{k}>0 for integer K. A related parameter
is H = Dh and H≤min(,
1).
We will refer to points (x_{i}, t^{n}) as boundary points if
i = 0 or N, or if n = 0. Similarly, we refered to them as interface points if
x = and
n>0. Otherwise, they are interior points.
We defined a function u(x, t) at mesh point (x_{i}, t^{n}). Let u_{i}^{n} = u(x_{i}, t^{n}). Also we defined two difference operators:
The onedimensional finite difference domain decomposition algorithm designed
in Zuo and Yuan (2003) is as follows:
Algorithm 1
U_{i}^{n} = u_{i}^{n} at boundary points:
at interface points, n = 1, 2, ..., M
at interior points, n = 0, 1, 2, ..., M
at interface points, n = 0
where, U_{i}^{n} is the numerical solution of the exact solution
of the problem (1). Algorithm 1 can be shown that the computation from time
level n1 to n is at first to solve the interface points by using Du FortFrankel
scheme and then to solve the interior points on the two subdomains (0, )
and (,
1) independently. For the solution U_{i}^{n} of Algorithm 1
it holds that:
Theorem 1: The truncation error of Algorithm 1 is:
It is generally required to limit the step site τ and h, when,
in the calculation to achieve the accuracy 0(Δt+h^{2}).
Theorem 2: Algorithm 1 is absolutely stable.
In this study, new domain decomposition algorithms are presented for the twodimensional
heat equation based on Algorithm 1 in Zuo and Yuan (2003).
The algorithms are unconditional stable and their truncation errors are:
New algorithms for the twodimensional heat equation: Consider the following twodimensional heat conduction equation, u(x, y, t) is a solution of the heat equation on Ω = (0,1)x(0, 1). Specifically, u satisfies:
We start a simple twodomain scheme. Take:
Let y_{1} = jh, j = 0, ..., N. In analogy with the onedimensional
case we will call a points (x_{i}, y_{j}, t^{n}) as
boundary points if n = 0 or if.
Such a point with x_{i} = will
be an interface point if it is not a boundary point. The remaining points (x_{i},
y_{j}, t^{n}) are in (Ω_{1}∪Ω_{2})x(0,
T] and are interior points. The values U^{n}_{i,j} will approximate
.
Defined the difference operators:
We extended the onedimensional parabolic equation to twodimensional one. At interface points we use the improved Du FortFrankel scheme; at interior points we use the fullimplicit scheme. The algorithm is as follows:
Algorithm 2:
Algorithm 3:
Algorithm 4:
Stability and truncation error analysis of the new algorithms: In order to verify the stability of the Algorithm 2, 3 and 4, we developd the following lemmas:
Lemma 1: (Li, 2004) if b and c are real numbers
and x_{1}, x_{2} are the roots of x^{2}bxc = 0,then
we have x_{i}<1, i = 1, 2 if and only if b#12<2.
Here, we only discuss the stability and error of Algorithm 4.
Theorem 3: Algorithm 4 is unconditional stable.
We only examine the difference scheme at interface points, the formula at interface points of Eq. 12 can be deformed into:
Transform it into its equivalent twolevel difference equations:
Let U = (U, V)^{T} and make to
substitute the above formula, then:
the growth matrix is:
the characteristic equation of G(τ, k) is:
by Lemma 1, the characteristic roots of G(τ, k) satisfy the following condition:
It is able to get the growth matrix G#1, thus proving that the algorithm is unconditionally stable.
Similar as the proof of Theorem 3, the Algorithms 2 and 3 are unconditionally stable.
Theorem 4: The truncation error of Algorithm 4 is:
Proof: Here, we analysis the truncation error of Eq. 12 at point (i, j, n):
Similar as Algorithm 4, the truncation errors of Algorithm 2 and 3 are:
it is generally required to limit the step site τ and h, when τ→0, (τ/H)→0, in the calculation to achieve the accuracy 0(Δt+h^{2}).
RESULTS
Let u°(x, y) = sin(πx)sin(πy) in the twodimensional heat Eq. 5. Then the exact solution of Eq. 5 is:
In Table 16, N =20, D = 2, t = 0.5, x
= 0.5 absolute error 1 denotes the absolute error between exact solution and
Algorithm 1, absolute error 2 denotes the absolute error between exact solution
and Algorithm 2, absolute error 3 denotes the absolute error between exact solution
and Algorithm 3, absolute error 4 denotes the absolute error between exact solution
and Algorithm 4.
Table 1 and 2 show the comparisons of the calculation results between Algorithm 2 and 1 when the grid ratio r is 0.4 and 8, respectively for two dimensional heat equation. From the numerical results we conclude that: if the value r≤2, the degrees of accuracy between two methods are comparable. While when the value r>2, the numerical solution of the Algorithm 1 begins to show the obvious instability. Algorithm 2 maintains the unconditional stability without degrading the overall approximation order.
Table 3 and 4 show the comparisons of the calculation results between Algorithm 3 and Algorithms 1 when the grid ratio r is 0.4 and 8 respectively for twodimensional heat equation. From the numerical results we conclude that: if the value r≤2, the degrees of accuracy between two methods are comparable. While when the value r>2, the numerical solution of the Algorithm 1 begins to show the obvious instability. Algorithm 3 not only maintains the unconditional stability, but also does not decrease the overall approximation order.
Table 1: 
The numerical results when h = 0.05, Δt = 0.0001, r
= 0.4 

Table 2: 
The numerical results when h = 0.05, Δt = 0.0002, r
= 8 

Table 3: 
The numerical results when h = 0.05, Δt = 0.0001, r
= 0.4 

Table 4: 
The numerical results when h = 0.05, Δt = 0.0002, r
= 8 

Table 5: 
The numerical results when h = 0.05, Δt = 0.0001, r
= 0.4 

Table 6: 
The numerical results when h = 0.05, Δt = 0.0002, r
= 8 


Fig. 1: 
The exact solution when h = 0.01, Δt = 0.01, y = 0.01 
The Table 5 and 6 show the comparisons of the calculation results between Algorithm 4 and Algorithm 1 when the grid ratio r is 0.4 and 8, respectively for two dimensional heat equation. Based on the numerical results, the following conclusions are drawn: if the value r≤2, the degrees of accuracy between two methods are comparable. When the value r>2, the numerical solution of the Algorithm 1 begins to show the obvious instability. Algorithm 4 maintains both the unconditional stability and the overall approximation order.
The real solution and approximate solution obtained from Algorithm 2 are shown in Fig. 1 and 2, respectively. The experimental results are obtained under the following conditions: h is 0.01; Δt is 0.01; y is 0.01.
From Fig. 1 and 2, we find that the approximate solution of Algorithm 2 is comparable with the exact solution of Algorithm 1. The truncation error of Algorithm 2 is:

Fig. 2: 
The approximate solution when h = 0.01, Δt = 0.01, y
= 0.01 
Moreover, this algorithm is absolute stable. This verifies the superiority of our algorithm.
CONCLUSION
In this study, we developd a novel domain decomposition method with better stability and calculation accuracy, while we obtain the stability condition and the maximum norm error estimates.
In the twodimensional case, through integrating the interface, we use Du FortFrankel scheme to establish a series of domain decomposition methods. Those methods improve the stability condition when compared to the other methods which are proposed in the previous study.
Analysis results show that appropriately integrating the interface, particularly, the use of Du FortFrankel scheme can realize the domain decomposition, meanwhile, can greatly improve the stability conditions of the algorithm without decreasing the overall approximation order of the algorithm.
ACKNOWLEDGMENTS
This research was also supported by Natural science fund for colleges and universities in Jiangsu Province (08KJD520018), Natural Science Foundation from Nanjing University of Information and Science Technology (20080302).