On Convergence of Difference Schemes for IBVP for Quasilinear Parabolic Equations with Generalized Solutions

For initial boundary value problem (IBVP) for onedimensional quasilinear parabolic equations with generalized solutions an usual linearized difference scheme is constructed. The uniform parabolicity condition 0 < k 1 ≤ k(u) ≤ k 2 is assumed to be fulfilled for the sign alternating solution u(x, t) ∈ ¯ D(u) only in the domain of exact solution values (unbounded nonlinearity). On the basis of the proved new corollaries of the maximum principle not only two sided estimates for the approximate solution y but its belonging to the domain of exact solution values are established. We assume that the solution is continuous and its first derivative ∂u ∂x has discontinuity of the first kind in the neighborhood of the finite number of discontinuity lines. An existence of time derivative in any sense is not assumed. Convergence of approximate solution to generalized solution of differential problem in the grid norm L 2 is proved.