HOMOGENEOUS DIFFERENCE SCHEMES ON NON-UNIFDRM NETS FOR EI)UATIONS OF PARABOLIC TYPE*

The basic problems of the theory of homogeneous difference schemes for linear, quasi-linear aud non-linear equations of parabolic type have been studied in a number of works ([ll- 151). The stability, convergence, and also estimates of the rate of convergence (order of accuracy) of several families of homogeneous difference schemes in the classes of continuous aud ~s~tinu ~ coefficients of the differential equation have been established. In [sl attention was paid to the fact that on an arbitrary sequence of non-uniform nets difference schemes which have second order approximation on uniform nets have only first order approxi-mation. For this reason the problem of the order of accuracy on non-uui-form nets requires special study. A family of homogeneous difference schemes on non-uniform nets for the equation L’k. 90 0,- iz (w$)-!?(5)U+m =o (14 was given in Es]. The definition of this family was the sBme as in the case of uniform nets with h~ogeneous difference schemes, studied in [?I. It was shown that the order of accuracy of these schemes on non-uniform nets is equal to the order of their accuracy on uniform nets both in the case of continuous coefficients in the differential equation and in the case of discontinuous coefficients. It is interesting to note that this result is obtained if we use a priori estimates of the ssme type as in the case of discontinuous coefficients [71 end a uniform net. l%e effective characteristic of the net is the mean square ste