A priori estimates for the solution of the difference analogue of a parabolic differential equation

Abstract

When finite difference methods are used to solve a differential equation one of the most important theoretical problems which arises is to determine the convergence of the difference scheme in question when the difference net is divided up in an unrestricted way. The difference z between the solution y of the difference boundary problem and the solution u of the corresponding problem for the original differential equation usually satisfies a non-homogeneous difference equation with homogeneous boundary and initial conditions. The right-hand side ψ of this equation denotes the approximation error for the difference scheme on the solution u of the original problem. The question of the convergence of the difference scheme reduces to the estimate of the function z in the following form. ∥ z∥ 1 ⩽ M ∥ ψ∥ 2, (a) where ∥∥ 1 and ∥∥ 2 are norms and M is a positive constant independent of the difference net. Work devoted to a priori estimates for simple difference approximations of parabolic type differential equations has appeared recently [1], [2]. A particular interest is attached to a priori estimates when there are difference schemes for which the principle of the maximum does not, in general, hold good. In the study of the convergence of difference schemes in the class of smooth coefficients, the asymptotic orders of approximation and accuracy usually coincide, i.e. ∥ z∥ 1 ⩽ M ∥ ψ∥ 1, where ‖z‖ 1= max ω h |z| and ω h is the difference net. For discontinuous coefficients this is not generally true (see [3], [4], [5]). In the neighbourhood of a discontinuity of the coefficients the difference operator, generally speaking, does not approximate to the differential operator. Therefore we cannot apply the principle of the maximum in the study of convergence. Other a priori estimates with a specially selected norm must be found. In the article [3], using the example of schemes for the very simple equation L (k, q, f)u = d dx [k (x) du dx ] − q (x) u + tf (x) = 0 it was shown that convergence in the class of discontinuous coefficients follows from the a priori estimate of the form (a), where ‖ψ‖ 2= ∑ i=1 N−1 h ∑ k=1 i hψ k . An a priori estimate was obtained in [1] on the assumption that the difference analogue of the heat conduction coefficient was “differentiable” with respect to x. In Section 1 we derive a similar a priori estimate free from this restriction, and this enables us to use the estimate for stationary (motionless) discontinuities of the heat conduction coefficient as well. We shall consider the difference boundary problem with boundary conditions of a very general form. In Section 2 we obtain an integral formula enabling us to obtain a priori estimates on the assumption only that the net functions, the coefficients of the equation, are bounded. These estimates are valid for the axisymmetric case and for the spherically-symmetric case. The new a priori estimates are an effective means for proving the convergence also of the estimate of the accuracy of the difference schemes in the class of discontinuous coefficients. However we shall consider questions of convergence and accuracy of the various difference schemes separately.