B.A.V. Bennett and M.D. Smooke,

Numerical Linear Algebra with Applications, 8, 513–536, 2001

Abstract

The local rectangular refinement (LRR) solution-adaptive gridding method automatically produces orthogonal unstructured adaptive grids and incorporates multiple-scale finite differences to discretize systems of elliptic governing partial differential equations (PDEs). The coupled nonlinear discretized equations are solved simultaneously via Newton's method with a Bi-CGSTAB linear system solver. The grids' unstructured nature produces a nonstandard sparsity pattern within the Jacobian. The effects of two discretization schemes (LRR multiple-scale stencils and traditional single-scale stencils) on Jacobian bandwidth, convergence speed, and solution accuracy are studied. With various point orderings, for two simple problems with analytical solutions, the LRR multiple-scale stencils are seen to: 1) produce Jacobians of smaller bandwidths than those resulting from the traditional single-scale stencils; 2) lead to significantly faster Newton's method convergence than the single-scale stencils; and 3) produce more accurate solutions than the single-scale stencils. The LRR method, including the LRR multiple-scale stencils, is finally applied to an engineering problem governed by strongly coupled, highly nonlinear PDEs: a steady-state lean Bunsen flame with complex chemistry, multicomponent transport, and radiation modeling. Very good agreement is observed between the computed flame height and previously published experimental data.