On convergence of difference schemes for Dirichlet IBVP for two-dimensional quasilinear parabolic equations

Abstract

For Dirichlet initial boundary value problem (IBVP) for two-dimensional quasilinear parabolic equations, a monotone linearised difference scheme is constructed. The uniform parabolicity condition 0 < k1 ≤ kα(u) ≤ k2, α = 1, 2 is assumed to be fulfilled for the sign alternating solution 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 derivatives have discontinuities of the first kind in the neighbourhood of the finite number of discontinuity lines. No smoothness of the time derivative is assumed. Convergence of approximate solution to generalised solution of differential problem in the grid norm L2 is proved.