On the convergence and accuracy of homogeneous difference schemes for one-dimensional and multidimensional parabolic equations

Abstract

Homogeneous difference schemes, the general definition of which is given in [I], were considered from the point of view of their application to equations of the parabolic type with one space variable in [21-151. Since the problem of the convergence of difference schemes can be reduced to the problem of the stability of the solution of a linear equation with respect to its right hand side, and to the boundary and initial data, a priori estimates were obtained in the first instance in [21 and [41 from which the stability follows. As in [II, special attention was paid to the choice of norms for the estimation of the right hand side of the difference equation with the help of which the convergence of homogeneous schemes could be proved in the class of discontinuous coefficients of the differential equation. In [31 the a priori estimates obtained in [2l were used in the proof of the uniform convergence and in the estimate of the order of accuracy of homogeneous difference schemes for the linear equation of heat conduction with discontinuous coefficients. In [51 homogeneous schemes were studied for a non-linear equation (1) of the parabolic type with boundary conditions of kind III.