NUMERICAL SOLUTION OF THE DIRICHLET PROBLEM FOR A STATIONARY NONLINEAR SYSTEM OF PARTIAL DIFFERENTIAL EQUATIONS
Abstract
This paper presents a numerical solution to the Dirichlet problem applied to a stationary nonlinear system of partial differential equations. The methodology integrates the pseudo-time stepping method with finite difference schemes, utilizing the backward sweep algorithm (Thomas algorithm) and the method of alternating directions (MAD). Comprehensive numerical experiments validate the proposed framework, demonstrating second-order spatial convergence and robust stability across highly coupled nonlinear domains. Furthermore, the study shows that the judicious selection of initial approximations is critical for accelerating steady-state convergence and mitigating computational overhead in strongly nonlinear scenarios.
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