Fully explicit finite-difference methods for two-dimensional diffusion with an integral condition
Introduction
Heat conduction and other diffusion processes with an integral condition replacing one boundary condition arises in many important applications in heat transfer [1], [2], [3], [9], [11], [15], [17], control theory [16], thermoelasticity [5], [6] and medical science [4].
The purpose of this article is to present very efficient explicit second- and fourth-order-accurate finite-difference methods for solving the following two-dimensional time-dependent diffusion equation:with initial conditionand boundary conditionsand the integral conditionwhere f, g0, g1, h0, h1, s and m are known functions, while the functions u and μ are unknown.
The boundary condition (6) is variable separable, with spatial dependence given by h0(x) and time dependence given by μ(t).
The existence and uniqueness of the solution of this non-classic problem has been studied in [3]. The organization of this paper is as follows:
Numerical schemes for the solution of , , , , , , are described in Section 2. The method of incorporating (7) with μ unknown is described in Section 3, and numerical results for various test cases produced by the methods developed, are given in Section 4.
In each case error estimates are given in the maximum norm. Section 5 concludes this paper with a brief summary.
Section snippets
The numerical solution with Dirichlet boundary conditions
We divide the domain [0,1]2×[0,T] into M2×N mesh with spatial step size h=1/M in both x and y directions and the time step size k=T/N, respectively. Grid points (xi,yj,tn) are given bywhere M is an even integer. We use ui,jn and μn to denote the finite-difference approximations of u(ih,jh,nk) and μ(nk), respectively. The numerical methods suggested here are based on two ideas: Firstly, the standard second-order FTCS method, or the 9-point
The numerical integration procedure
In this section we describe the numerical integration procedure, used for (7) employing Simpson's rule which has a truncation error of order four.
The presence of an integral term in a boundary condition can greatly complicate the application of standard numerical techniques [8]. The accuracy of the quadrature must be compatible with that of the discretization of the differential equation [7]. In the following the fourth-order Simpson's composite “one-third” rule is used.
Consider the integral
Numerical tests
Two problems for which exact solutions are known are now used to test the methods described. Firstly, these are applied to solve , , , , , with μ(t) given, in order to test the two methods used to compute values of ui,jn+1 from ui,jn, in the interior of the solution domain. Secondly, these are applied to solve , , , , , , with μ(t) not given a priori, thereby testing the algorithm used for the two-dimensional time-dependent problem with an integral condition replacing one boundary condition.
Conclusion
In this paper three fully explicit finite-difference schemes, the standard FTCS method and the 9-point finite-difference scheme and the (1,13) fully explicit technique were applied to the two-dimensional diffusion equation with an integral condition replacing one-boundary condition. The latter worked very well for two-dimensional diffusion with an integral condition, because of its fourth-order accuracy. These methods seems particularly suited for parabolic partial differential equations with
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