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

Abstract

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 ≤ k ( u ) ≤ k 2 $0<k_1\le k(u) \le k_2$ is assumed to be fulfilled for the sign alternating solution u ( x , t ) ∈ D ¯ ( u ) $u(x,t) \in \bar{D}(u)$ 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 ∂ u ∂ x $\frac{\partial u}{\partial x}$ 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 L 2.