On an L-stable method for stiff differential equations

Bui, Tien D.

doi:10.1016/0020-0190(77)90014-x

Cited by 14 publications

(5 citation statements)

References 14 publications

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“…To avoid order reduction, extra order conditions are required, which are implemented in this paper to develop the new algorithm. Also, it has been shown [26] that the extra condition to resolve order reduction contradicts the L-stability of the Rosenbrock method, so there is no L-stable Rosenbrock method that is also free of order reduction. The new method can be used to solve nonlinear parabolic partial differential equation with all types of boundary conditions; however, for Neumann or Robin boundary condition, extra efforts are needed to form the semidiscrete ODE system.…”

Section: Resultsmentioning

confidence: 99%

A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation

Liao

2015

Abstract and Applied Analysis

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The semidiscrete ordinary differential equation (ODE) system resulting from compact higher-order finite difference spatial discretization of a nonlinear parabolic partial differential equation, for instance, the reaction-diffusion equation, is highly stiff. Therefore numerical time integration methods with stiff stability such as implicit Runge-Kutta methods and implicit multistep methods are required to solve the large-scale stiff ODE system. However those methods are computationally expensive, especially for nonlinear cases. Rosenbrock method is efficient since it is iteration-free; however it suffers from order reduction when it is used for nonlinear parabolic partial differential equation. In this work we construct a new fourth-order Rosenbrock method to solve the nonlinear parabolic partial differential equation supplemented with Dirichlet or Neumann boundary condition. We successfully resolved the phenomena of order reduction, so the new method is fourth-order in time when it is used for nonlinear parabolic partial differential equations. Moreover, it has been shown that the Rosenbrock method is strongly A-stable hence suitable for the stiff ODE system obtained from compact finite difference discretization of the nonlinear parabolic partial differential equation. Several numerical experiments have been conducted to demonstrate the efficiency, stability, and accuracy of the new method.

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Section: Resultsmentioning

confidence: 99%

A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation

Liao

2015

Abstract and Applied Analysis

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“…is the minimized SSD per segment and Y a1 and Y a2 are the average values of y in the respective segments. The Rosenbrock procedure, which does not require the solution of nonlinear equations, has been investigated in [38][39][40]. The implementation of this procedure requires only the solution of linear systems of algebraic equations, a much simpler task compared to the first two approaches.…”

Section: Methodsmentioning

confidence: 99%

Identifying Unwanted Conditions through Chaotic Area Determination in the Context of Indonesia’s Economic Resilience at the City Level

2021

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The purpose of this research is to determine the unwanted condition as a strategic criterion in measuring the economic resilience of a city. A new approach in determining economic resilience was developed to overcome the weaknesses of the index method commonly used internationally. Based on the output of this research, the development priority program for each city becomes distinctive depending on the status of the city’s economic resilience. Quality improvement programs are used for cities that do not have resilience and retention programs for cities that already have economic resilience. Five piecewise linear regression parameters are applied to identify a statistical model between Income per capita and Pc as a concern variable and modifier variable, and a Z. Model is tested massively involving all 514 cities in Indonesia from 2015 to 2019, covering the components of the modifier variable: local revenue (PAD), poverty, unemployment and concern variable; GRDP and population. The value of the Fraction of variance unexplained (FVU) of the model is 40%. This value is obtained using the Rosenbrock Pattern Search estimation method with a maximum number of iterations of 200 and a convergence criterion of 0.0001. The FVU area is a condition of uncertainty and unpredictability, so that people will avoid this area. This condition is chaotic and declared as an unwanted condition. The chaotic area is located in the value of UZ less than IDR 5,097,592 and Pc < Pc (UZ) = 27,816,310.68, and thus the coordinates of the chaotic boundary area is (5,097,592: 27,816,310.68). FVU as a chaotic area is used as the basis for stating whether or not a city falls into unwanted conditions. A city is claimed not to be economically resilient if the modifier variable Z is in a chaotic boundary.

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“…Dessa forma, a condição para L-estabilidade requer que o parâmetro 𝑑 tenha um dos seguintes valores: 𝑑 1 = 3,100316735$, 𝑑 2 = 0,5728160625, 𝑑 3 = 0,2204284103 ou 𝑑 4 = 0,1064387921. Ao usar o valor de 𝑑 2 para 𝑑, Bui (1977) desenvolveu um método L-estável de quarta ordem. Esse método está definido pelo conjunto de parâmetros descritos na Tabela 1. com os parâmetros da Tabela 1 e de um outro método A-estável de quarta ordem com quatro estágios, desenvolvido por Bui (1979b), está ilustrada na Fig.…”

Section: Implementaçãounclassified

MÉTODO NUMÉRICO PARA SOLUÇÃO DE EDOs RÍGIDAS NA MODELAGEM DE REAÇÕES QUÍMICAS

Sehnem¹,

Quadros²,

Buske³

2018

MUNDI ETG

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Problemas envolvendo reações químicas possuem uma característica importante denominada rigidez, indicando que as soluções dos sistemas de EDOs envolvidos variam em diferentes ordens de grandeza. Isso faz com que se escolham métodos numéricos adequados que tenham o intuito de obter soluções numéricas que sejam estáveis e convergentes a um custo computacional reduzido. Os métodos mais eficientes para tratar esse tipo de problema são os métodos implícitos, pois possuem uma região de estabilidade ilimitada no plano complexo que permite grandes variações no tamanho do passo. Neste trabalho propomos um método L-estável: o método de Rosenbrock de quarta ordem com quatro estágios. Simulam-se três problemas envolvendo reações químicas e os resultados apresentam boa concordância com a literatura.

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On an L-stable method for stiff differential equations

Cited by 14 publications

References 14 publications

A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation

A Strongly A-Stable Time Integration Method for Solving the Nonlinear Reaction-Diffusion Equation

Identifying Unwanted Conditions through Chaotic Area Determination in the Context of Indonesia’s Economic Resilience at the City Level

MÉTODO NUMÉRICO PARA SOLUÇÃO DE EDOs RÍGIDAS NA MODELAGEM DE REAÇÕES QUÍMICAS

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