Numerical methods for non-linear parabolic boundary-value problems with a priory bounded solution

We present a new family of numerical methods to solve non-linear parabolic boundary-value problems with constraints a priory imposed on the solution. The proposed procedures are based on a consistent first-order approximation of "diffusion" and "transport" terms combined with an unconditionally stable Gauss-Seidel-type iterative technique. We demonstrate that the proposed algorithms provide an overall priority with regard to conventional numerical schemes as applied to a diffusion wave model of open flows and to a Richards-type model of saturated-unsaturated flows.