Abstract.We construct an usual linearized difference scheme for initial boundary value problems (IBVP) for one-dimensional quasilinear parabolic equations with generalized solutions.
The uniform parabolicity condition $0<k_1\le k(u) \le k_2$ is assumed to be
fulfilled for the sign alternating solution $u(x,t) \in \bar{D}(u)$
only in the domain of exact solution values (unbounded
non-linearity). On the basis of new corollaries of the
maximum principle, we establish not only two-sided estimates for the approximate
solution y but its belonging to the domain of exact solution
values. We assume that the solution is continuous
and its first derivative $\frac{\partial u}{\partial x}$ 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. We prove convergence of the approximate solution to
the generalized solution of the differential problem in the grid norm L2.
