The effect of overall discretization scheme on Jacobian structure, convergence rate, and solution accuracy within the local rectangular refinement method

The local rectangular refinement (LRR) solution-adaptive gridding method au tomatically produces orthogonal unstructured adaptive grids and incorporate s multiple-scale finite differences to discretize systems of elliptic gover ning partial differential equations (PDEs). The coupled non-linear discreti zed equations are solved simultaneously via Newton's method with a Bi-CGSTA B linear system solver. The grids' unstructured nature produces a nonstanda rd 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 resulti ng from the traditional single-scale stencils; (2) lead to significantly fa ster Newton's method convergence than the single-scale stencils; and (3) pr oduce more accurate solutions than the single-scale stencils. The LRR method, including the LRR multiple-scale stencils, is finally appli ed to an engineering problem governed by strongly coupled, highly non-linea r PDEs: a steady-state lean Bunsen flame with complex chemistry, multicompo nent transport, and radiation modeling. Very good agreement is observed bet ween the computed flame height and previously published experimental data. Copyright (C) 2001 John Wiley & Sons, Ltd.