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@MastersThesis{Zarzur:2018:MéIMIn,
               author = "Zarzur, Antonio Maur{\'{\i}}cio",
                title = "M{\'e}todos IMEX para integra{\c{c}}{\~a}o temporal da 
                         Equa{\c{c}}{\~a}o de Burgers",
               school = "Instituto Nacional de Pesquisas Espaciais (INPE)",
                 year = "2018",
              address = "S{\~a}o Jos{\'e} dos Campos",
                month = "2018-04-27",
             keywords = "diferen{\c{c}}as finitas, equa{\c{c}}o\̃,es diferenciais 
                         parciais, esquemas IMEX, integra{\c{c}}a\̃,o temporal, 
                         modelagem nume\́,rica, finite differences, IMEX schemes, 
                         partial differential equations, time integration, numerical 
                         modeling.",
             abstract = "Simula{\c{c}}{\~o}es computacionais baseiam-se em modelos 
                         matem{\'a}ticos desenvolvidos para certas classes de 
                         fen{\^o}menos. A solu{\c{c}}{\~a}o computacional de 
                         equa{\c{c}}{\~o}es diferenciais parciais requer a escolha de um 
                         m{\'e}todo de integra{\c{c}}{\~a}o temporal capaz de simular, 
                         de forma est{\'a}vel, a evolu{\c{c}}{\~a}o do problema. H{\'a} 
                         m{\'e}todos mais adequados para determinadas classes de 
                         fen{\^o}menos, n{\~a}o existindo um m{\'e}todo geral que sirva 
                         adequadamente para todos os fen{\^o}menos. Deve-se levar em conta 
                         a acur{\'a}cia e a estabilidade do m{\'e}todo adotado, bem como 
                         seu desempenho computacional. De forma geral, os m{\'e}todos de 
                         integra{\c{c}}{\~a}o temporal s{\~a}o classificados como 
                         impl{\'{\i}}citos ou expl{\'{\i}}citos. Cada fam{\'{\i}}lia 
                         apresenta vantagens e desvantagens na solu{\c{c}}{\~a}o de 
                         determinadas classes de problemas. Uma abordagem mais recente, 
                         denominada IMEX, visa combinar as vantagens de cada 
                         estrat{\'e}gia para solucionar equa{\c{c}}{\~o}es com escalas 
                         de tempo vari{\'a}veis, de forma que os termos r{\'a}pidos 
                         s{\~a}o resolvidos implicitamente e os mais lentos s{\~a}o 
                         resolvidos explicitamente. O resultado {\'e} uma 
                         combina{\c{c}}{\~a}o de diferentes esquemas que otimiza o tempo 
                         de processamento ao evitar passos de tempo desnecessariamente 
                         pequenos para os termos r{\'a}pidos. Este trabalho prop{\~o}e a 
                         aplica{\c{c}}{\~a}o dessa abordagem na solu{\c{c}}{\~a}o da 
                         equa{\c{c}}{\~a}o de Burgers viscosa, objetivando realizar um 
                         estudo de caso, analisando sua acur{\'a}cia e desempenho 
                         computacional. A equa{\c{c}}{\~a}o de Burgers {\'e} uma das 
                         equa{\c{c}}{\~o}es fundamentais da din{\^a}mica de fluidos e 
                         serve como uma simplifica{\c{c}}{\~a}o das equa{\c{c}}{\~o}es 
                         de Navier-Stokes sem a presen{\c{c}}a da equa{\c{c}}{\~a}o de 
                         continuidade e dos gradientes de press{\~a}o, possuindo assim 
                         diversas aplica{\c{c}}{\~o}es pr{\'a}ticas. Sua 
                         solu{\c{c}}{\~a}o exata {\'e} conhecida, o que permite a 
                         compara{\c{c}}{\~a}o da acur{\'a}cia dos m{\'e}todos IMEX 
                         propostos com os tradicionais esquemas impl{\'{\i}}citos ou 
                         expl{\'{\i}}citos. Experimentos num{\'e}ricos demonstram que os 
                         m{\'e}todos propostos produzem solu{\c{c}}{\~o}es com o mesmo 
                         grau de acur{\'a}cia dos m{\'e}todos tradicionais, ao mesmo 
                         tempo em que estendem as condi{\c{c}}{\~o}es de estabilidade 
                         al{\'e}m dos limites de m{\'e}todos puramente 
                         expl{\'{\i}}citos. ABSTRACT: Computational simulations are based 
                         on mathematical models developed for certain phenomena. The 
                         numerical solution of partial differential equations requires the 
                         choice of a method for time integration capable of stably 
                         simulating the evolution of a problem. There are methods that are 
                         more suitable to certain classes of phenomena, and therefore no 
                         single, general method can be applied to every problem. Both 
                         accuracy and stability of the chosen method must be taken into 
                         account, as well as its computational efficiency. Generally 
                         speaking, time integration schemes are categorized as either 
                         implicit or explicit. Each of these broad families presents pros 
                         and cons when solving certain classes of problems. A more modern 
                         approach, called IMEX, seeks to combine the advantages of each 
                         strategy to solve equations containing both fast and slow 
                         time-scales in a way that the slow terms can be solved explicitly, 
                         while the slow terms are solved implicitly. This results in a 
                         combination of different schemes with the goal of optimizing 
                         processing time by avoiding unnecessarily small time steps for the 
                         fast terms. This dissertation utilizes this approach in solving 
                         the viscous Burgers equation as a test case for such methods, 
                         analyzing their accuracy and computational performance. Burgers 
                         equation is one of the fundamental equations in fluid dynamics 
                         which essentially simplifies the Navier-Stokes equations by 
                         removing the pressure gradient terms and continuity equation, thus 
                         serving diverse practical applications. Because its exact solution 
                         is known, comparisons can be drawn between the accuracy of the 
                         proposed IMEX schemes and that of more traditional implicit or 
                         explicit schemes. Numerical experiments are performed to 
                         demonstrate their ability to simulate the problem with the same 
                         order of accuracy achieved by traditional means, while extending 
                         the stability conditions beyond the reach of purely explicit 
                         schemes.",
            committee = "Campos Velho, Haroldo Fraga de (presidente) and Stephany, Stephan 
                         (orientador) and Freitas, Saulo Ribeiro de (orientador) and Rosa, 
                         Reinaldo Roberto and Dias, Pedro Leite da Silva",
         englishtitle = "IMEX Methods for time integration of Burgers' equation",
             language = "pt",
                pages = "91",
                  ibi = "8JMKD3MGP3W34R/3R25SCH",
                  url = "http://urlib.net/rep/8JMKD3MGP3W34R/3R25SCH",
           targetfile = "publicacao.pdf",
        urlaccessdate = "24 nov. 2020"
}


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