Splitting-methods based on Approximate Matrix Factorization and Radau-IIA formulas for the time integration of advection diffusion reaction PDEs

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• 作者：S. Gonz&aacute ; lez-Pinto ^{spinto@ull.es" class="auth_mail" title="E-mail the corresponding author} ; D. Hern&aacute ; ndez-Abreu ; ^{dhabreu@ull.edu.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Evolutionary advection&ndash ; diffusion-reaction partial differential equations ; Method of lines ; Approximate Matrix Factorization ; Runge&ndash ; Kutta Radau IIA methods ; Finite differences ; Stability and convergence
• 刊名：Applied Numerical Mathematics
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：104
• 期：Complete
• 页码：166-181
• 全文大小：531 K

文摘

A family of splitting methods for the time integration of evolutionary Advection Diffusion Reaction Partial Differential Equations (PDEs) semi-discretized in space by Finite Differences is obtained. The splitting is performed in the Jacobian matrix by using the Approximate Matrix Factorization (AMF) and by considering up to three inexact Newton Iterations applied to the two-stage Radau IIA method along with a very simple predictor. The overall process allows to reduce the storage and the algebraic costs involved in the numerical solution of the multidimensional linear systems to the level of 1D-dimensional linear systems with small bandwidths.

Some specific AMF-Radau methods are constructed after studying the expression for the local error in semi-linear equations, and their linear stability properties are widely studied. The wedge of stability of the methods depends on the number of splittings used for the Jacobian matrix of the spatial semidiscretized ODEs, class="mathmlsrc">ce" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0168927415001464&_mathId=si1.gif&_user=111111111&_pii=S0168927415001464&_rdoc=1&_issn=01689274&md5=0ae734e82f0f6cfb76a0e4fd57e35682">class="imgLazyJSB inlineImage" height="20" width="92" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0168927415001464-si1.gif">cript>cal-align:bottom" width="92" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0168927415001464-si1.gif">cript>class="mathContainer hidden">class="mathCode">, where h   stands for the spatial grid resolution. A-stability is proven for the cases class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0168927415001464&_mathId=si162.gif&_user=111111111&_pii=S0168927415001464&_rdoc=1&_issn=01689274&md5=7a0bc686fe3cfd2fa2cc5af9b96d84dd" title="Click to view the MathML source">d=1,2class="mathContainer hidden">class="mathCode">, and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0168927415001464&_mathId=si232.gif&_user=111111111&_pii=S0168927415001464&_rdoc=1&_issn=01689274&md5=058a9cc66a9641f09a9624400ad614e4" title="Click to view the MathML source">A(0)class="mathContainer hidden">class="mathCode">-stability for any class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0168927415001464&_mathId=si233.gif&_user=111111111&_pii=S0168927415001464&_rdoc=1&_issn=01689274&md5=55cd97fb0cc1131658b42cfae2835f1c" title="Click to view the MathML source">d≥1class="mathContainer hidden">class="mathCode">.

Numerical experiments on a 3D semi-linear advection diffusion reaction test problem and a 2D-combustion model are presented. The experiments show that the methods compare well with standard classical methods in parabolic problems and can also be successfully used for advection dominated problems when some diffusion or stiff reactions are present. In the latter case the stability imposes restrictions on the number of splitting terms (d).