A Particular Solution Trefftz Method for Non-linear Poisson Problems in Heat and Mass Transfer

Abstract

This paper proposes a computational procedure based on the Trefftz method for the solution of non-linear Poisson problems. The problem is solved by finding an approximate particular solution to the Poisson equation and using boundary collocation to solve the resulting Laplace equation. This method results in a global collocation and hence eliminates the need for discretization of the domain in both two and three dimensions. The solution procedure proposed earlier in the literature based upon the above method has been reformulated for increased computational efficiency. A quasi-Newton iteration method along with a new heuristic for source point location is used for efficient convergence of the numerical scheme. The efficacy of the new formulation has been demonstrated for two classes of problems, viz. the thermal explosion problem and the diffusion–reaction problem in a partially wetted catalyst pellet. A brief error analysis of the method, as well as future research directions, is presented.