Alternating direction method for two-dimensional parabolic equation with nonlocal integral condition ∗

Received: 28 December 2011 / Revised: 15 February 2012 / Published online: 24 February 2012 Abstract. Two-dimensional parabolic equation with nonlocal condition is solved by alternating direction method in the rectangular domain. Values of the solution on the boundary points are bind with the integral of the solution in whole two-dimensional domain. Because of this nonlocal condition, the classical alternating direction method is complemented by the solution of low dimension system of algebraic equations. The peculiarities of the method are considered.

Motivation and possible applications of the problem are indicated in [1].Parabolic equations with nonlocal conditions of different types at present are an intensively considered field both in the theory of differential equations and in numerical analysis.In papers [2][3][4][5][6][7][8][9][10], linear and nonlinear two-dimensional parabolic equations with nonlocal conditions are solved by the finite difference method.
The particularity of our problem (1)-( 5) under consideration is that the value of the solution in nonlocal condition (4) at the boundary points is linked with a two-dimensional integral of the solution.This is the difference of our research from the analogous researches in the above mentioned articles [2][3][4][5][6][7][8][9][10].Solution of two-dimensional parabolic equation with a nonlocal condition of type (4) by the finite difference method has considered rather in little.In [11], equation ( 1) with f = 0 and the integral condition.
is solved in rectangle domain by the finite difference method under the assumption Condition ( 4) is a partial case of condition (6).
The main result of our paper is the fact that we show that two-dimensional parabolic equations with a nonlocal condition of type ( 4) can be successfully solved by an efficient alternating direction method, and that condition (7) is not always necessary for this purpose.

Statement of a difference problem
Let us introduce the notation: Let us write the Peaceman-Rachford alternating direction method [12] for a differential problem (1)- (5). and Formula ( 12) is a trapezoidal rule for a two-dimensional integral.The quantity g n+1 i0 in this formula is defined as follows Note that problem ( 10)-( 12) differs from the usual one-dimensional difference problems of the alternating direction method.Namely, we cannot solve system (10) for a single fixed value i = 1, 2, . . ., N − 1 separately -in nonlocal condition (12) interconnected.

Algorithm for solving difference equations
We present an algorithm how to find u n+1 ij when the values of u n ij are known.The first part of algorithm is problem (8)-( 9) realized by the classical Thomas algorithm, namely, we have to solve a system of difference equations with a three-diagonal matrix for each value of the index j separately, j = 1, 2, . . ., N − 1.
In the second part of algorithm we have to solve system (10)-( 12) witch, as mentioned above, due to condition (12), cannot be solved separately with a single fixed value of the index i.Therefore, we solve this system (10)-( 12) by a modified Thomas algorithm, described in [6], in which the algorithm was applied to the system of difference equations with a nonlocal condition, simpler than condition (12).
First of all we write system of equation ( 10) in the form where According to the modified Thomas algorithm, first of all we write the solution of system ( 10)- (12) to each fixed value i = 1, 2, . . ., N − 1 in the form Note that in the general case, the coefficient αj should also depend on indices i and n, however, regarding the specificity of equation ( 10) (coefficients B, C in equation ( 14) do not depend on i, j and n), the coefficient αj in formula (15) will not depend on other indices.Using equation ( 14) and condition (11), we calculate by the classical Thomas algorithm: Further we use theoretically the Thomas algorithm once more for each i = 1, 2, . . ., N − 1, but we look for u n+1 ij in the following form where α 0 = 1, β n+1 i0 = 0. From expressions ( 15) and ( 18) we derive We require now that solution (18) would satisfy nonlocal condition (12).By substituting expressions (18) into condition (12), we obtain For each value of the index i = 1, 2, . . ., N − 1.In expressions (21) the quantities u n+1 i0 , i = 1, 2, . . ., N − 1 are unknown.These values are found by solving the system of equations (21) which we rewrite in the following shape: where Thus, in order to realize the second part of the algorithm, i.e. to solve the system of equations ( 10)-( 12) with a nonlocal condition, first, we need to find the coefficients αj , βn+1 ij by the Thomas algorithm, and then to calculate the coefficients α j , β n+1 ij by formulas ( 19), (20).To achieve this aim, the number of arithmetic operations is proportional to N 2 , i.e. proportional to the number of unknowns in one layer.Afterwards we have to solve the system of (N − 1)-order linear algebraic equations (22).To this end, the number of arithmetic operations is proportional to N 3 (Gausian elimination) or N 2 (iterative methods).After finding u n+1 i0 , we have only to make use of formula ( 18).Let us consider some main properties of the system of equations ( 22).Lemma 1.For each τ > 0 and h > 0 there exists a strict estimate Proof.Taking into consideration the values of coefficients B and C, we get from formula ( 16) Regarding the condition α 0 = 1, it follows from formula (19) that These estimates and formula (23) directly yield the proposition of the lemma.
is true.According to the assumption of the lemma on the function γ(x), we get Hence follows that det A > 0.
Remark.If |γ(x)| 2, the system of equations ( 22) can be solved by a stable algorithm.

Numerical results
Problem ( 1)-( 5) with the function γ(x) = ce x , choosing different values of c, has been solved by the method described in this paper.The problem with γ(x) = c was solved as well.In this case, in fact there is no need to solve the system of equation ( 22), because it follows from the condition γ(x) = c that u n+1 i0 with all the values of i is an unknown constant, so the system of equations ( 22) degenerates to a single equation.Expressions of the functions f (x), ϕ(x) and µ i (x), i = 1, . . ., 4 were selected so that the function u * (x, y, t) = sin(πx) sin(πy)e 2t were an exact solution of problem ( 1)- (5).
Numerical results are written in Tables 1-2.The errors r = max i,j u * x i , y j , t n − u n ij as t n = T are presented in these tables.One of the main goals of the numerical experiment was to obtain information on the stability of a difference scheme.As far as it is known, when solving a parabolic equation with any type of nonlocal conditions by the finite difference method, one of the most important things is that the stability of a difference scheme depends on the parameters or functions presented in nonlocal conditions.
The authors of papers [3,9,13] have proved that while considering nonlocal conditions of different types in one or two-dimensional case, the sufficient stability condition of a difference scheme in a certain energetic norm can be where S is a transition matrix of the difference scheme expressed in the shape The structure of the spectrum of the matrix S for difference scheme ( 8)-( 12) considered by us has not yet been analyzed, therefore we could observe the fact of stability or instability only from the numerical result.Note that the spectrum of difference operator with nonlocal conditions can be very complicated [14,15].
In addition we can compare our numerical results with the theoretical result on the stability of a difference scheme for a one-dimensional parabolic equation that formally corresponds to problem (1)-( 5): The stability condition of finite difference scheme for this problem is [13]: which has a connection (at random or naturally) with the matrix property det A > 0 of system (22) (see Lemma 2. Table 1 indicates, that the numerical results, in the example solved by us, allow us to think that there is a certain analogy between problems (1)-( 5) and ( 22) on the subject of stability.