Some Stable Explicit Difference Approximations to the Diffusion Equation

Abstract

1. Introduction. The use of digital computing machines in finding approximate solutions of partial differential equations is extensive. Because of expense in using such machines, there is considerable motivation to find the most efficient means of solution. Consider the one-dimensional form of the diffusion equation in Cartesian coordinates. If the time derivative is approximated by a forward difference and the distance derivative is approximated by a central difference at the original time level, the system is explicit. That is, the dependent variable at the succeeding time step, may be written explicitly in terms of that variable at the earlier time step. It is well known [5] that stability requirements limit the size of the time step in proportion to the square of the size of the distance increment. For many applications the allowable time step is prohibitively small. By employing implicit methods, this limitation is removed [5]. In these methods the distance derivative is approximated by a central difference formula, but evaluated at the succeeding time step. Solution of simultaneous equations is necessary to establish values of the dependent variable at the new time level. It appears that the scheme of Saul'ev combines the best features of the two methods mentioned above. It avoids solving simultaneous equations but retains stability for any size time step.For simplicity, the method of Saul'ev will be applied to the diffusion equation

Numerical methods for conjunctive two-dimensional surface and three-dimensional sub-surface flows

“…The moisture diffusivity coefficient h i (q) in Equation (8) is discretized as given in Table II. Among the five methods, three of them are of the explicit type: purely explicit, ADE of Barakat and Clark [19], and ADE of Larkin [20]. The other two are of the implicit type: purely ADI and ADI of Brian [21].…”

“…Five of the six numerical methods applied here are explicit schemes: the purely explicit method, Saul'yev's [22] downstream method, Saul'yev's upstream method, Larkin's [20] ADE method, and Barakat and Clark's [19] ADE method. The ADI method applied here has an implicit alternating direction scheme and is identical with the Peaceman and Rachford [23] ADI method.…”

“…For the two ADE methods and the ADI method, theoretical analyses showed that the methods are unconditionally stable for the linear two-dimensional heat diffusion equation [15,19,20]. The present study investigates the stability characteristics of the six numerical methods for the non-linear heat diffusion equation as in Equation (1).…”

“…Figure 14(a) shows the relation between the normalized error and time increment for the five numerical methods investigated, in which the values of i were calculated every 20 s from 20 to 400 s, i.e. m= 20 in Equation (20). The ADE method of Barakat and Clark, ADE method of Larkin, ADI method of Brian, and the purely ADI method have almost the same accuracy.…”

Solution of the Space-Dependent Reactor Kinetics Equations in Three Dimensions

“…18 In addition, an exponential transformation is applied to Eq. (1.6), which greatly improves the truncation error.…”