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The coupled non\u2010linear discretized equations are solved simultaneously via Newton's method with a Bi\u2010CGSTAB linear system solver. The grids' unstructured nature produces a nonstandard sparsity pattern within the Jacobian.<\/jats:p><jats:p>The effects of two discretization schemes (LRR multiple\u2010scale stencils and traditional single\u2010scale 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\u2010scale stencils are seen to: (1) produce Jacobians of smaller bandwidths than those resulting from the traditional single\u2010scale stencils; (2) lead to significantly faster Newton's method convergence than the single\u2010scale stencils; and (3) produce more accurate solutions than the single\u2010scale stencils.<\/jats:p><jats:p>The LRR method, including the LRR multiple\u2010scale stencils, is finally applied to an engineering problem governed by strongly coupled, highly non\u2010linear PDEs: a steady\u2010state 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. Copyright \u00a9 2001 John Wiley &amp; Sons, Ltd.<\/jats:p>","DOI":"10.1002\/nla.265","type":"journal-article","created":{"date-parts":[[2002,8,25]],"date-time":"2002-08-25T16:20:51Z","timestamp":1030292451000},"page":"513-536","source":"Crossref","is-referenced-by-count":3,"title":["The effect of overall discretization scheme on Jacobian structure, convergence rate, and solution accuracy within the local rectangular refinement method"],"prefix":"10.1002","volume":"8","author":[{"given":"Beth Anne V.","family":"Bennett","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mitchell D.","family":"Smooke","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2001,11,5]]},"reference":[{"key":"e_1_2_1_2_2","doi-asserted-by":"publisher","DOI":"10.1006\/jcph.1993.1012"},{"key":"e_1_2_1_3_2","doi-asserted-by":"publisher","DOI":"10.1080\/00102209508907812"},{"key":"e_1_2_1_4_2","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1620370308"},{"key":"e_1_2_1_5_2","unstructured":"de LangeHC.Modelling of premixed laminar flames. 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