ScienceAsia 27 (2001) : 051-062 |doi: 10.2306/scienceasia1513-1874.2001.27.051

New Finite-Difference Schemes for Constrained Non-Linear Parabolic Equations with Application to the Porous Medium Flows

Stanislav S Makhanov^*

ABSTRACT: We present a new family of numerical methods to solve non-linear parabolic initial boundary value problems with constraints imposed a priori on the solution. Firstly, we introduce our recent results developed for the diffusion wave equation endowed with one constraint (non-negativity of the solution). The numerical procedures are based on a consistent first-order approximation of cdiffusione and ctransporte terms combined with the Gauss-Seidel-type iterative technique. Secondly, we show that the Richard’s type models of porous medium flows exemplify the general case of a non-linear parabolic equation endowed with the bilateral constraints specified by characteristics of the porous medium. Therefore, we generalize the preceding numerical procedures to the Richard’s type models of unsaturated flows in the soil. The Gauss-Seidel-type iterative technique is supplemented by a three-step numerical procedure employing auxiliary variables. We prove the convergence theorems and introduce some further1 extensions of the algorithm. Finally, we verify the proposed schemes by methodological applications and analyze the convergence rate.

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Department of Information Technology, Sirindhorn International Institute of Technology, Thammasat University, Rangsit Center, Patum Thani 12121, Thailand.

* Corresponding author, E-mail: makhanov@siit.tu.ac.th

Received 27 Apr 2000, Accepted 8 Dec 2000