ࡱ> 130AbjbjUU >$??A  0 %%%%%%%%%%%%`")%%m0%%%%H%%%%%%%%%%%%%%%%%%%%%%%%% : Equaes Diferenciais so frequentemente utilizadas na modelagem de fenmenos fsicos. Diversas abordagens podem ser empregadas na resoluo aproximada dessas equaes. Entre essas tem-se os modelos aproximativos discretos (Mtodos das Diferenas Finitas, dos Volumes Finitos, dos Elementos Finitos, etc). Na resoluo dos problemas aproximados podem-se utilizar estratgias computacionais como decomposio de operadores (ex: Mtodo de direes alternadas, IMEX, Hopscotch, etc). A decomposio espao-temporal Hopscotch permite a computao eficiente das equaes do modelo aproximado, devido s atualizaes temporais explcita e implcita alternadas em conjuntos complementares de pontos. As frmulas de diferenciao regressiva (Backward Differentiation Formulas - BDF) so utilizadas para gerar mtodos de passo mltiplo para equaes diferenciais ordinrias (ou de equaes diferenciais parciais discretizadas com o mtodo das linhas). Os mtodos baseados em discretizao ao longo das linhas caractersticas, como o Mtodo das Caractersticas Modificado, permitem adequao ao comportamento de equaes evolutivas em referenciais lagrangianos. Esse recurso fornece uma combinao promissora com as tecnologias de computao contemporneas.Neste trabalho um novo mtodo numrico introduzido, visando tanto o aumento da ordem de aproximao quando comparado ao Hopmoc, como tambm fornecer um mtodo numrico eficiente para uso em computadores paralelos com memria compartilhada. O Mtodo BDF-Hopmoc um mtodo implcito semilagrangeano, como o Mtodo Modificado das Caractersticas; emprega a decomposio de pontos da grade com hopscotch ao longo de linhas caractersticas, como no Hopmoc; Alm disso, o BDF-Hopmoc emprega uma abordagem de vrias etapas ao longo do tempo (caracterstico) por meio de frmulas BDF. Neste trabalho, a consistncia e a estabilidade so comprovadas para o BDF-Hopmoc unidimensional e bidimensional com o auxlio da anlise de von Neumann. Comparaes entre os resultados obtidos com o mtodo Hopmoc e o mtodo BDF-Hopmoc de segunda ordem (BDF2) em problemas com dominncia convectiva so apresentadas. O Mtodo BDF-Hopmoc aplicado a problemas da dinmica de fluidos computacionais. Eficincia e preciso so investigadas para problemas envolvendo as Equaes de Burgers. O BDF-Hopmoc implementado em um computador paralelo. Uma nova mtrica proposta para enriquecer comparaes entre mtodos implcitos convencionais e Hopscotch. Palavras-chave: mtodo Hopscotch; mtodo das caractersticas modificado; anlise da convergncia de von Neumann; equaes de Burgers; frmulas de diferenciao regressivas; mtodos numricos para equaes diferenciais parciais. Differential Equations are often used as mathematical models for the investigation of physical phenomena. Several computational approaches can be used to solve these equations. Discrete approximate models include Finite Difference Methods, Finite Volumes, Finite Elements, etc.. For the solution of the approximate problems, in particular for high-performance computing, strategies such as operator decomposition (some examples are Modified Method of Characteristics, IMEX, Hopscotch, etc.) can be used. The Hopscotch space-time decomposition allows the efficient computation of the approximate model's equations. This is due to the explicit and implicit alternate time updates in complementary sets of points, since all computation is performed without the need for solving linear systems. It has long been established that, for convection dominated problems such as fluid flows, the discretization of the differential equations along the lines of flow, namely the lagrangian approach, provides better stability features than the corresponding eulerian discretizations. One of the first lagrangian techniques developed was the Modified Method of Characteristics (MMOC), which gives rise to an implicit along-the-flow method. Backward Differentiation Formulas (BDF) are implicit multistep methods that may be used to provide high order time approximations (or along the direction of the independent variable in the method of lines), especially for stiff initial value problems' solvers. When one considers the issue of performance, explicit methods lead to faster computations than their implicit counterparts. This led to the inception of the Hopmoc method, which allows one to have an implicit lagrangian method exclusively through explicit calculations. This feature provides a promising match with contemporary computer technologies. In this work a new numerical method is introduced, aiming at both the increase of the order of approximation when compared with Hopmoc, as well as providing an efficient numerical method for use in shared memory parallel computers. The BDF-Hopmoc Method is a semilagrangian implicit method, like the Modified Method of Characteristics; it employs the decomposition of grid points for hopscotching along characteristic lines, as in Hopmoc; additionally, BDF-Hopmoc employs a multistep approach along (the characteristic) time through BDF formulas. In this work, consistency and staability are proven for the one-dimensional and two-dimensional BDF-Hopmoc with the aid of von Neumann's analysis. Comparisons between results obtained with the Hopmoc Method and the second order BDF-Hopmoc Method (BDF2) in problems with convective dominance are presented. BDF-Hopmoc Method is applied to computational fluid dynamics problems. Efficiency and accuracy are investigated for problems involving Burgers' Equations. BDF-Hopmoc is implemented on a parallel computer. A new metric is proposed in order to enrich comparisons between conventional implicit methods and hopscotching-implicit ones. Keywords: Hopscotch method; modified method of characteristics; von Neumann convergence analysis; Burgers' equation; backward differentiation formulas; numerical methods for partial differential equations. Qir9Q^s{Ahy h26]mH sH h2mH sH hy h2mH sH hy h26] hy h2h2qrsA$a$gdy <P1h:pF|. 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