On the structure of errors for Radau IA methods applied to index-2 DAEs

Aubry, Anne; Chartier, Philippe

doi:10.1016/s0168-9274(96)00023-2

Cited by 5 publications

(2 citation statements)

References 15 publications

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“…Radau methods are A-stable [5] but expensive to implement like all the implicit methods that need some sort of Newton iteration [17], [6], [5]. Some alternatives have been proposed to reduce the computational cost that are usually based on approximate Jacobian matrix and dimensionality reduction [17].…”

Section: Radau Methodsmentioning

confidence: 99%

“…Classical numerical methods both implicit and explicit [4], [5], [6], [7] can be employed to find accurate results because numerical integration of the non linear differential equations is performed. The transient stability analysis (TSA) is very important for the design of electric power systems because it allows to determine the system's ability to withstand large disturbances and to transition to a normal operating condition.…”

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Numerical Methods Coupled withRichardson Extrapolation for Computationof Transient Power Systems

et al. 2017

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The numerical solution of transient stability problems is a key element for electrical power system operation. The classical model for multi-machine systems is defined as a set of non-linear differential equations for the rotor speed and the generator angle for each electrical machine, this mathematical model is usually known as the swing equations. This paper presents how to use direct Richardson extrapolation of several orders for the numerical solution of the swing equations and compares it with other commonly used implicit and explicit solvers such as Runge-Kutta, trapezoidal, Shampine and Radau methods. A numerical study on a simple three machine system is used to illustrate the performance and implementation of algebraic Richardson extrapolation coupled to several solution methods. Normally, Métodos numéricos acoplados con la extrapolación de Richardson para el cálculo de sistemas de potencia transitoriosResumen La solución numérica de problemas de estabilidad transitoria es un elemento clave para la operacion de sistemas eléctricos. El modelo clásico para sistemas multi-máquina se define como un conjunto de ecuaciones diferenciales no lineales para la velocidad del rotor y el ángulo del generador para cada máquina eléctrica, este modelo matemático se conoce generalmente como las ecuaciones de oscilación. Este artículo presenta la forma de utilizar la extrapolación directa de Richardson de varios órdenes para la solución numérica de las ecuaciones de oscilación y la compara con otros métodos implícitos y explícitos de uso común como los métodos RungeKutta, Trapezoidal, Shampine y Radau. Se presenta un estudio numérico sobre un sistema simple de tres máquinas para ilustrar el desempeño y la implementación algebráica de la extrapolación de Richardson. El orden de exactitud de cualquier solución numérica puede aumentarse cuando se utiliza la extrapolación de Richardson. Se proporciona un ejemplo numérico para una red eléctrica que consta de tres máquinas y nueve buses que sufren una perturbación. Se demuestra que en este caso la extrapolación de Richardson aumenta efectivamente el orden de exactitud de los métodos explícitos haciéndolos competitivos con los métodos implícitos Palabras clave: Estabilidad transitoria de sistemas de potencia; extrapolación de Richardson; ecuaciones dinámicas.

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Section: Radau Methodsmentioning

confidence: 99%

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Numerical Methods Coupled withRichardson Extrapolation for Computationof Transient Power Systems

et al. 2017

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High‐order implicit Runge–Kutta time integrators for fluid‐structure interactions

Cori

Étienne

Garon

et al. 2015

Numerical Methods in Fluids

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SummaryThis paper presents an approach to develop high‐order, temporally accurate, finite element approximations of fluid‐structure interaction (FSI) problems. The proposed numerical method uses an implicit monolithic formulation in which the same implicit Runge–Kutta (IRK) temporal integrator is used for the incompressible flow, the structural equations undergoing large displacements, and the coupling terms at the fluid‐solid interface. In this context of stiff interaction problems, the fully implicit one‐step approach presented is an original alternative to traditional multistep or explicit one‐step finite element approaches. The numerical scheme takes advantage of an arbitrary Lagrangian–Eulerian formulation of the equations designed to satisfy the geometric conservation law and to guarantee that the high‐order temporal accuracy of the IRK time integrators observed on fixed meshes is preserved on arbitrary Lagrangian–Eulerian deforming meshes. A thorough review of the literature reveals that in most previous works, high‐order time accuracy (higher than second order) is seldom achieved for FSI problems. We present thorough time‐step refinement studies for a rigid oscillating‐airfoil on deforming meshes to confirm the time accuracy on the extracted aerodynamics reactions of IRK time integrators up to fifth order. Efficiency of the proposed approach is then tested on a stiff FSI problem of flow‐induced vibrations of a flexible strip. The time‐step refinement studies indicate the following: stability of the proposed approach is always observed even with large time step and spurious oscillations on the structure are avoided without added damping. While higher order IRK schemes require more memory than classical schemes (implicit Euler), they are faster for a given level of temporal accuracy in two dimensions. Copyright © 2015 John Wiley & Sons, Ltd.

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