The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae

Cash, J. R.

doi:10.1016/0898-1221(83)90122-0

Cited by 128 publications

(66 citation statements)

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“…Portanto, a metodologia é considerada A(α)-estável para ordens menores ou iguais a nove e Rígida-estável para ordens maiores que nove. Estas propriedades de estabilidade são consideravelmente melhores que as atingidas pelos métodos baseados nas fórmulas BDF convencionais (Cash, 1983;Cash e Considine, 1992;Cash, 2000).…”

Section: Método De Diferenciação Regressiva Modificado Estendidounclassified

“…As técnicas numéricas abordadas neste trabalho são: Método de Diferenciação Regressiva Modificado Estendido Implícito (Cash, 1983), Método de Diferenciação Regressiva Modificado Estendido com Esparsidade (Abdulla et al, 2000), Mé-todo de Diferenciação Regressiva (Petzold, 1983) e RungeKutta 4 a ordem. Na literatura esses métodos são identificados como MEBDF, MEBDFSD, BDF e RK4.…”

Section: Introductionunclassified

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Técnicas de solução de sistemas de equações diferenciais e algébricas: aplicação em sistemas de energia elétrica

Pessanha¹,

Portugal²,

Paz³

2005

Sba Controle & Automação

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The present work investigates and compares the computational performance of numerical techniques applied to the solution of algebraic and differential equations (DAEs) representing power systems and their components. Among the considered techniques, emphasis was given to the method known as Modified Extended Backward Differentiation Formulae (MEBDF), due to its numerical stability properties that conventional Backward Differentiation Formulae (BDF) methods do not have, and these properties may avoid numerical troubles during DAEs solution. The comparison is made through computational tests related to simulations of power system short-and long-term stability phenomena (transient angular stability and long-term voltage stability), using IEEE system-test. The main objective was to check the computational efficiency of these numerical techniques. The computational aspect is related to the simulation CPU time. técnicas numéricas aplicadas na solução de Equações Diferenciais e Algébricas (EDAs) representando sistemas elé-tricos e seus componentes. Entre os métodos considerados, destaca-se o método conhecido como Método de Diferenciação Regressiva Modificado Estendido (MEBDF), por este apresentar propriedades de estabilidade numérica que os Mé-todos de Diferenciação Regressiva (BDF) convencionais não apresentam, sendo que estas propriedades podem evitar problemas numéricos durante a solução das EDAs. As técni-cas são comparadas através de testes envolvendo simulações computacionais no domínio do tempo de fenômenos de estabilidade de curta-, e de longa-duração (angular e de tensão, respectivamente) em sistemas-teste do IEEE. O objetivo principal foi verificar a eficiência dessas técnicas numéri-cas sob o ponto de vista computacional, sem comprometer a precisão. O aspecto computacional está relacionado com o tempo de cpu gasto nas simulações.

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Section: Método De Diferenciação Regressiva Modificado Estendidounclassified

Section: Introductionunclassified

Técnicas de solução de sistemas de equações diferenciais e algébricas: aplicação em sistemas de energia elétrica

Pessanha¹,

Portugal²,

Paz³

2005

Sba Controle & Automação

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“…The k-step MEBDF-methods are A-stable [109] for k ≤ 3, while for BDF this is restricted to the case k ≤ 2 [33].. Thus these MEBDF-methods 'break' Dahlquist's Law [109] that applies to real multistep methods: we have higher order methods with unconditional stability.…”

Section: T) By Expanding F(ẏ(t) X(t) T)mentioning

confidence: 99%

“…Cash proposed the Modified Extended BDF (MEBDF) method, which combines better stability properties and higher order of convergence than BDF, but requires more computations per step [33,34]. One timestep with the MEBDF method consists of three BDF steps and an evaluation step.…”

Section: T) By Expanding F(ẏ(t) X(t) T)mentioning

confidence: 99%

Modelling and Discretization of Circuit Problems

Günther

Feldmann

Maten

2005

Handbook of Numerical Analysis

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“…Both subcategories, multistage Runge-Kutta and linear multistep methods, can be fullyimplicit (the current time-level is obtained by solving a nonlinear problem that uses information from the current time-step) and fully-explicit (the current time-level is calculated using information coming from the previous time-steps only). Representative classes of time-integration schemes embedded in the GL method consist of implicit multistep methods such as Adams-Moulton (AM) [22] and backward differentiation (BDF) methods [13,20,21], implicit multistage Runge-Kutta schemes such as diagonally (DIRK) and singly-diagonally (SDIRK) implicit Runge-Kutta schemes [3,19,59], explicit multistep methods, such as leapfrog and Adams-Bashforth methods [28,43], explicit Runge-Kutta schemes, such as the fourthorder Runge-Kutta scheme [55] and partitioned methods, such as Implicit-Explicit (IMEX) schemes, whereby the operators are linearized in some fashion with-e.g., two Butcher tableaux, one explicit and one implicit [5,40,106]. While EBTI schemes are widely used in computational fluid dynamics, especially in the engineering sector [18,52], their adoption in the weather and climate communities has been less widespread, with SE schemes [54,88,107] and horizontally-explicit vertically-implicit schemes [8,40,63]-i.e., schemes where the horizontal direction is treated explicitly and the vertical is treated implicitly-becoming more prominent but still confined mainly to research and limited-area models (with very few exceptions-see Table 1).…”

Section: Eulerian-based Time-integration (Ebti)mentioning

confidence: 99%

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

Mengaldo

Wyszogrodzki

Diamantakis

et al. 2018

Arch Computat Methods Eng

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The continuous partial differential equations governing a given physical phenomenon, such as the Navier-Stokes equations describing the fluid motion, must be numerically discretized in space and time in order to obtain a solution otherwise not readily available in closed (i.e., analytic) form. While the overall numerical discretization plays an essential role in the algorithmic efficiency and physically-faithful representation of the solution, the time-integration strategy commonly is one of the main drivers in terms of cost-to-solution (e.g., time-or energy-to-solution), accuracy and numerical stability, thus constituting one of the key building blocks of the computational model. This is especially true in time-critical applications, including numerical weather prediction (NWP), climate simulations and engineering. This review provides a comprehensive overview of the existing and emerging time-integration (also referred to as time-stepping) practices used in the operational global NWP and climate industry, where global refers to weather and climate simulations performed on the entire globe. While there are many flavors of time-integration strategies, in this review we focus on the most widely adopted in NWP and climate centers and we emphasize the reasons why such numerical solutions were adopted. This allows us to make some considerations on future trends in the field such as the need to balance accuracy in time with substantially enhanced time-to-solution and associated implications on energy consumption and running costs. In addition, the potential for the co-design of time-stepping algorithms and underlying high performance computing hardware, a keystone to accelerate the computational performance of future NWP and climate services, is also discussed in the context of the demanding operational requirements of the weather and climate industry.

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The integration of stiff initial value problems in ODEs using modified extended backward differentiation formulae

Cited by 128 publications

References 5 publications

Técnicas de solução de sistemas de equações diferenciais e algébricas: aplicação em sistemas de energia elétrica

Técnicas de solução de sistemas de equações diferenciais e algébricas: aplicação em sistemas de energia elétrica

Modelling and Discretization of Circuit Problems

Current and Emerging Time-Integration Strategies in Global Numerical Weather and Climate Prediction

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