Homogeneous difference schemes for non-linear parabolic equations

Abstract

We shall consider homogeneous difference schemes of “through” computation (see [1]–[3]) for the non-linear parabolic equation Pu = ∂ ∂x (k(x,t) ∂u ∂x ) + (x,t,u, ∂u ∂x , ∂u ∂t ) = 0 with a discontinuous “heat-conduction coefficient” k( x, t). Attention is devoted chiefly to determining the order of accuracy of the six-point schemes P α hτ (see [3]) for the third boundary problem in a bounded region g ̄ P(0⩽x⩽1,0⩽t⩽T) . The investigation is carried out directly for a wide class of homogeneous difference schemes (see § 1.4), given in terms of pattern functionals of the type indicated in [1] and [3]. The functionals satisfy conditions which ensure a second order of approximation (with respect to x) of the scheme. Similar difference schemes (in the case of the first boundary problem) for the linear heat-conduction equation when (x,t,u, ∂u ∂x , ∂u ∂t ) = (x,t)−q(x,t)u−(x,t) ∂u ∂t were studied in [3]. Difference boundary conditions of the third kind, with the same order of approximation as the scheme in the class of solutions of the equations Pu = 0 are formulated in § 2. The problem of the accuracy of the difference problem obtained can be reduced to estimating the solution z of a linear difference equation with linear difference boundary conditions and zero initial condition by using the functions ψ, v 1 and v 2, where ψ is the approximation error of the scheme, v 1 and v 2 are the approximation errors of the boundary conditions in the class of solutions of the equation Pu = 0. A priori estimates are used to evaluate the solution of this problem. These are generalized from the estimates [4] and [5] to a more general equation and more general conditions. The so-called “fixed discontinuities” of the coefficient k( x, t), i.e. discontinuities on the finite number of straight lines x = η v = const., parallel to the t-axis in the plane ( x, t), are considered in this paper. By separating the error connected with the approximation error of the boundary conditions it can be shown that the third boundary problem has the same order of accuracy as the first (see § 3). The principal result has been formulated as Theorem 4 in § 3. It has been proved that any scheme P α hτ of the initial family (see § 1.4) converges uniformly for 0.5 ⩽ ⩽ α ⩽ 1, when the intervals of the difference set h = Δx and τ = Δt tend to zero independently. The order of accuracy in the class of discontinuous coefficients is also evaluated. Results for the explicit schemes ( α = 0) are not given because they can be formulated by analogy with the case of the linear equation (3). As has been pointed out several times (see [1], [3], [5]), the difficulty in investigating uniform convergence in the case of the discontinuous functions k( x, t) and ( x, t, u, p, q) lies in the fact that the difference scheme does not approximate to a differential operator near the line of discontinuity of the coefficient k( x, t). For six-point schemes convergences can be proved only by using an improved a priori estimate (Theorem 2) which uses the norms ¦|ψ¦| 5 and ||ψ|| 5. of a special type for an integral estimate of the approximation error ψ. Although uniform nets are considered, the principal results and, in particular, Theorem 4 are applicable to difference schemes on non-uniform nets also. This question has been treated separately. The method of investigation adopted can be used to prove Theorem 4 for more general boundary conditions, including non-linear ones also. § 1 and 3 remain practically unchanged in that case. It should be noted that there are a number of papers (for example [6]–[11]) devoted to the investigation of difference schemes of a partial type for non-linear and quasi-linear parabolic equations (1) in the class of smooth functions k( x, t) and ( x, t, u, p, q). In most of the papers either average convergence or uniform convergence has been studied using the principle of the maximum (involving boundedness for r/h2) for partial cases of eqn. (1). The method closest to ours is the one described in paper [10], in which the uniform convergence of a scheme for the case of boundary conditions of the first kind and the continuous functions k(x, t) andj(x, t, u, p, q) was proved. The results given there follow from Theorem 3 of our paper. The a priori estimates used in [10] are unsuitable for proving convergences in the class of discontinuous coefficients.