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Doctoral Thesis
DOI
10.11606/T.55.2008.tde-08052008-144555
Document
Author
Full name
Vanessa Avansini Botta
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Carlos, 2008
Supervisor
Committee
Barcelos, Celia Aparecida Zorzo
Dimitrov, Dimitar Kolev
Mckee, James Clark Saint Clair Sean
Menegatto, Valdir Antonio
Meneguette Junior, Messias
Title in Portuguese
Zeros de polinômios característicos e estabilidade de métodos numéricos
Keywords in Portuguese
Estabilidade de métodos numéricos
Métodos (K, L)
Order stars
Zeros de polinômios característicos
Abstract in Portuguese
A Teoria das equações diferenciais faz parte de uma área da Matemática muito rica em aplicações. Os métodos numéricos para a solução de equações diferenciais ordinárias são, da mesma forma que as próprias equações, fontes importantes de problemas a serem pesquisados. Como destaque tem-se os métodos multiderivadas de passo múltiplo, que são importantes na solução de problemas stiff. Os métodos numéricos mais conhecidos para a solução desses problemas são os BDF, que compõem, para L = 1, a família dos métodos (K, L) de Brown. Algumas questões relacionadas à estabilidade dos métodos (K, L) ainda não foram solucionadas como, por exemplo, uma conjectura de Jeltsch. Para analisá-la, é necessário estudar o comportamento dos zeros dos polinômios característicos associados aos métodos (K, L). Neste trabalho é apresentado um estudo sobre zeros de polinômios com o objetivo de demonstrar a validade da conjectura de Jeltsch para K '< OU =' 'K IND; L' . As regiões de estabilidade para alguns valores de K e L fixos são apresentadas e também é utilizada a teoria das order stars para mostrar algumas propriedades dos métodos (K, L). Portanto, este trabalho apresenta um estudo sobre os métodos (K, L) de Brown e usa uma ferramenta pouco utilizada na literatura, que são as order stars, para demonstrar alguns resultados
Title in English
Zeros of characteristic polynomials and stability of numerical methods
Keywords in English
Brown (K, L) methods
Order stars
stability of numerical methods
Zeros of characteristic polynomials
Abstract in English
THe theory of differential equations is part of one area of Mathematics very rich in applications. The numerical methods for the solutions of ordinary differential equations are, in the same way as the equations themselves, important sources of problems to be studied. As prominence one has the multiderivative multistep methods which are important for the solution of stiff problems. The best known numerical methods for the solutions of these kind of problems are the BDF methods, which is part of the family of the Brown (K,L) methods with L = 1. Some questions about stability of the (K, L) methods has not been solved yet as, for example, a conjecture by Jeltsch. In order to tackle this open problem, it becomes necessary to study the behavior of the zeros of the characteristic polynomials associated to the (K, L) methods. In this work a study of the zeros of the characteristic polynomial is carried out aiming at proving Jeltsch conjecture for K < OR = 'K IND.L'. Regions of stability is shown for some fixed values of K and L, as well as the use of order stars techniques are applied to show some properties of (K, L) methods. Therefore, this work presents a study of Brown's (K, L) methods, that makes use of a tool that seems not to have been used very often in the literature, the order stars, in order to prove the main results
 
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Publishing Date
2008-05-08
 
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