@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/ibi/8JMKD3MGP3W34R/3R25SCH",
targetfile = "publicacao.pdf",
urlaccessdate = "19 abr. 2024"
}