An optimal mesh choice in the numerical solution of the heat equation

We consider the one-dimensional heat conduction equation [1]. The so-called θ-method will be applied to the numerical solution of the problem [2,3]. Here, the question is the suitable choice of the mesh on which the continuous problem is discretized, that is, the choice of the stepsize of the discretization in both variables. The basic condition comes from the condition of the convergence [2,3]. Moreover, it is reasonable to choose such a convergent numerical method which is optimal in a certain sense [4,5]. For any given implicit finite difference method with fixed number of arithmetic operations, we introduce an optimal parameter choice and define the optimal mesh in this sense of values of the stepsizes. We compare our results with the bounds obtained for the preservation of basic qualitative properties. As a result, we obtain the parameter choice for any convergent method being both optimal and preserving the main qualitative properties. Finally, a numerical example is given. © 1999 Elsevier Science Ltd. All rights reserved.