Fully explicit finite-difference methods for two-dimensional diffusion with an integral condition

doi:10.1016/S0362-546X(00)00172-3

Nonlinear Analysis: Theory, Methods & Applications

Volume 48, Issue 5, February 2002, Pages 637-650

Professor John R. Cannon on the occasion of his 62nd birthday

https://doi.org/10.1016/S0362-546X(00)00172-3 Get rights and content

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: $∂u ∂t =α ∂^{2} u ∂x^{2} + ∂^{2} u ∂y^{2}$ with initial condition $u(x,y,0)=f(x,y), 0≤x,y≤1,$ and boundary conditions $u(0,y,t)=g_{0} (y,t), 0≤t≤T, 0≤y≤1,$ $u(1,y,t)=g_{1} (y,t), 0≤t≤T, 0≤y≤1,$ $u(x,1,t)=h_{1} (x,t), 0≤t≤T, 0≤x≤1,$ $u(x,0,t)=h_{0} (x)μ(t), 0≤t≤T, 0≤x≤1,$ and the integral condition $∫ 01 ∫ 0 s(x) u(x,y,t) d x d y=m(t), 0≤x,y≤1,$ where 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 (x_i,y_j,t_n) are given by $x_{i} =ih, i=0,1,2,…,M,$ $y_{j} =jh, j=0,1,2,…,M,$ $t_{n} =nk, n=0,1,2,…,N,$ where M is an even integer. We use u_i,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 u_i,jⁿ⁺¹ from u_i,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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View full text

Fully explicit finite-difference methods for two-dimensional diffusion with an integral condition

Introduction

Section snippets

The numerical solution with Dirichlet boundary conditions

The numerical integration procedure

Numerical tests

Conclusion

First page preview

J. Math. Anal. Appl.

J. Math. Anal. Appl.

J. Comput. Appl. Math.

J. Comput. Phys.

The solution of the diffusion equation in two-space variables subject to the specification of mass

Appl. Anal. J.

A reaction-diffusion system arising in modeling man-environment diseases

Quart. Appl. Math.

Existence of a property of solutions of the heat equation subject to linear thermoelasticity and other theories

Quart. Appl. Math.

A decreasing property of solutions of a parabolic equation with applications to thermoelasticity and other theories

Quart. Appl. Math.

The reformulation and numerical solution of certain nonclassical initial-boundary value problems

SIAM J. Sci. Stat. Comput.