Generalized integrating factor methods for stiff PDEs

Krogstad, Stein

doi:10.1016/j.jcp.2004.08.006

Cited by 208 publications

(166 citation statements)

References 18 publications

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“…Different types of exponential integrators exist. One classical approach is Lawson's method [23,22], where one starts by performing the variable transformation…”

Section: Definition 1 Consider the Equationmentioning

confidence: 99%

“…The parameter is typically small, imposing a high degree of stiffness in the system (2). A popular approach towards solving stiff systems in the form (1) has been the use of exponential integrators [5,12,22]. Such methods are motivated in part by computational efficiency considerations [13]; without sacrificing high-order accuracy, one gets rid of the severe restriction on the time step commonly associated with explicit methods for stiff problems.…”

Section: Introductionmentioning

confidence: 99%

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An exponential time-differencing method for monotonic relaxation systems

Aursand

Evje

Flåtten

et al. 2014

Applied Numerical Mathematics

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Abstract. We present first and second-order accurate exponential time differencing methods for a special class of stiff ODEs, denoted as monotonic relaxation ODEs. Some desirable accuracy and robustness properties of our methods are established. In particular, we prove a strong form of stability denoted as monotonic asymptotic stability, guaranteeing that no overshoots of the equilibrium value are possible. This is motivated by the desire to avoid spurious unphysical values that could crash a large simulation.We present a simple numerical example, demonstrating the potential for increased accuracy and robustness compared to established Runge-Kutta and exponential methods. Through operator splitting, an application to granulargas flow is provided.

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“…Different types of exponential integrators exist. One classical approach is Lawson's method [23,22], where one starts by performing the variable transformation…”

Section: Definition 1 Consider the Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An exponential time-differencing method for monotonic relaxation systems

Aursand

Evje

Flåtten

et al. 2014

Applied Numerical Mathematics

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“…In this paper we implement exponential integrators of the Runge-Kutta type. We consider the Runge-Kutta integrating factor (IFRK) [26,29,30], the Runge-Kutta exponential time differencing (ETDRK) [26,29], and the ET-DRK method with improved accuracy by Krogstad (ETDRKB) [31]. Further, we will use the numerically stable scheme by Kassam and Trefethen [26] for calculating the coefficients in the ETDRK methods.…”

Section: Exponential Integratorsmentioning

confidence: 99%

“…It is shown in [31] that the main step of Cox-Matthews method can be reproduced based on the techniques of continuous Runge-Kutta methods. Motivated from the same idea, but also applied to the internal stages of the method, a new fourth-order method is derived in [31].…”

Section: Runge-kutta Exponential Time Differencing (Etdrk)mentioning

confidence: 99%

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Fast and High Accuracy Numerical Methods for the Solution of Nonlinear Klein–Gordon Equations

Mohebbi

Asgari

Shahrezaee

2011

Zeitschrift Für Naturforschung A

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and a mohebbi@kashanu.ac.ir Z. Naturforsch. 66a, 735 -744 (2011) / DOI: 10.5560/ZNA.2011-0038 Received April 13, 2011 / revised July 6, 2011 In this work we propose fast and high accuracy numerical methods for the solution of the onedimensional nonlinear Klein-Gordon (KG) equations. These methods are based on applying fourthorder time-stepping schemes in combination with discrete Fourier transform to numerically solve the KG equations. After transforming each equation to a system of ordinary differential equations, the linear operator is not diagonal, but we can implement the methods such as for the diagonal case which reduces the time in the central processing unit (CPU). In addition, the conservation of energy in KG equations is investigated. Numerical results obtained from solving several problems possessing periodic, single, and breather-soliton waves show the high efficiency and accuracy of the mentioned methods.

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Comparison of sequential data assimilation methods for the Kuramoto–Sivashinsky equation

Jardak

Navon

Zupanski

2009

Numerical Methods in Fluids

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A new comparison of three frequently used sequential data assimilation methods illuminating their strengths and weaknesses in the presence of linear and nonlinear observation operators is presented. The ensemble Kalman filter (EnKF), the particle filter (PF) and the maximum likelihood ensemble filter (MLEF) methods were implemented and the spectral shallow water equations model in spherical geometry model was employed using the Rossby-Haurwitz Wave no. 4 test case as initial condition. Numerical tests conducted reveal that all three methods perform satisfactorily in the presence of linear observation operator for 10 to 15 days model integration, whereas the EnKF, even with the Evensen fixture [Evensen 03] for the nonlinear observation operator failed in all tested metrics. The particle filter PF and the hybrid filter MLEF both performed satisfactorily in the presence of highly nonlinear observation operators with a slight advantage in terms of CPU time to the MLEF method.

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Generalized integrating factor methods for stiff PDEs

Cited by 208 publications

References 18 publications

An exponential time-differencing method for monotonic relaxation systems

An exponential time-differencing method for monotonic relaxation systems

Fast and High Accuracy Numerical Methods for the Solution of Nonlinear Klein–Gordon Equations

Comparison of sequential data assimilation methods for the Kuramoto–Sivashinsky equation

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