Difference Scheme of Higher Order of Approximation for the Hallaire’s Equation with Variable Coefficients

Abstract

The initial boundary value problem for the one-dimensional Hallaire’s equation with variable coefficients and boundary conditions of the first kind is studied. The problem under study describes the processes of heat transfer in a heterogeneous environment, moisture transfer in soils, and fluid filtration in fractured porous media. To numerically solve the problem posed, a difference scheme of high order of accuracy is constructed: the fourth order of accuracy in h and the second order of accuracy in τ. Using the method of energy inequalities, an a priori estimate of the solution in a difference treatment is obtained. From this estimate it follows that the solution is unique and stable with respect to the right-hand side and initial data. Under the assumption of the existence of an exact solution to the original differential problem in the class of sufficiently smooth functions, and also due to the linearity of the problem under consideration, the obtained a priori estimate implies that the solution of the constructed difference problem converges to the solution of the original differential problem at a rate equal to the order of approximation of the difference scheme. The goal and scientific novelty of the work is to obtain a new numerical scheme of a higher order of approximation when solving the Dirichlet problem for the Hallaire’s equation with variable coefficients.