Difference Analogues of Two-Sided Estimates for Quasilinear Parabolic Equations with Mixed Derivatives and Arbitrary-Sign Reaction Coefficients

This paper deals with initial-boundary value problems (IBVPs) for quasilinear parabolic equations with unbounded nonlinearity, mixed derivatives, and reaction coefficients of arbitrary sign. Finite-difference schemes (FDSs) with second-order spatial approximation are constructed for these problems. It is proved that the proposed schemes satisfy a nonstandard maximum principle under separate conditions on the temporal and spatial grid steps, without any coupling restriction between them, thereby ensuring unconditional monotonicity. Moreover, using the Ladyzhenskaya transformation, discrete two-sided estimates for the numerical solution are derived and shown to be fully consistent with the corresponding estimates for the continuous problem. The convergence and stability of the proposed schemes are rigorously justified in the grid L 2 {L₂} -norm.