A linear three-level difference scheme for quasilinear parabolic equations

The purpose of this paper is to establish a convergence theorem for a second order accurate, linear, three-level difference scheme for a class of quasilinear parabolic differential equations.In the interests of notational simplicity, we shall present our result in detail only for the special semilinear partial differential equation (U) Mtt)*,¿(aWg).Later, we shall indicate how our convergence theorem can be extended to more general equations than (la).Let id = w(x, t) be a smooth solution of the differential equation in the rectangu-where (p, H and H sire specified functions.In addition to assuming that the coefficient functions b and a are defined and smooth for all real w, we assume that there exist positive constants p and n such that(2) b(w) §(i>0 and a(w) ê 17 > 0.Because of (2), it follows that the differential equation (la) is uniformly parabolic, so that vo is the only solution of the boundary value problem ( 1 ).Before describing the three-level difference scheme for (1), we introduce some convenient notation.If iV is a positive integer and A > 0 is defined by the relation