Two linearized second-order block-centered finite difference methods for nonlinear Sobolev equations

In this paper, two efficient, linearized (or semi-implicit) Crank-Nicolson block-centered finite difference algorithms for the strongly nonlinear Sobolev equations are investigated and analyzed. Newton linearization and linear extrapolation techniques are considered to treat the nonlinear terms. It is shown that on general nonuniform rectangular grids, under local Lipschitz continuous assumptions on the nonlinear coefficients and reaction terms, second-order temporal and spatial convergence are achieved for the primal scalar variable p, its gradient u and its flux q simultaneously. Stability of the presented algorithms are then rigorously proved under a rough time-step condition t = o(h1/2), where t and h are, respectively, the temporal and spatial mesh sizes. Numerical experiments are presented to show the efficiency and convergence of the proposed methods.