Efficient adaptive step size control for exponential integrators

Deka, Pranab J.; Einkemmer, Lukas

doi:10.1016/j.camwa.2022.07.011

Cited by 7 publications

(12 citation statements)

References 35 publications

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“…The conclusions that can be drawn from the results shown here are not completely in agreement with the results for the Kelvin-Helmholtz instability in Deka and Einkemmer [11]. In that work, we find Leja to be superior to Krylov for the EXPRB43 integrator.…”

Section: Exprb43contrasting

confidence: 67%

“…In Deka and Einkemmer [11], we reported better computational performance for the Krylov-based method, whilst maintaining high accuracy, for EPIRK5P1 in the case of magnetic reconnection. However, it is to be noted that in that work, we used a standard Krylov iterative scheme (not the highly advanced KIOPS method).…”

Section: Epirk5p1mentioning

confidence: 77%

“…This step size controller, or the cost controller from here on, has been optimised to incur minimum possible computational cost by choosing a smaller step size, than the one chosen by the traditional controller. The cost controller was first proposed for implicit integrators [14] and was later shown to be even more beneficial for exponential integrators [11,12]. This is because exponential integrators, in principle, are able to take much larger step sizes than implicit ones.…”

Section: Automatic Step Size Controlmentioning

confidence: 99%

“…In Deka and Einkemmer [11], we studied the performance of the Leja and Krylov-based methods for the set of equations of magnetohydrodynamics (MHD). We found that Leja had better performance than the Krylov method for the EXPRB43 [6,20,19] integrator whereas Krylov had surpassed Leja for the "Krylov-friendly" EPIRK5P1 [34] integrator.…”

Section: Introductionmentioning

confidence: 99%

“…2, we provide a gist of the Leja and Krylov methods as well as our variable step size implementation for improved computational performance, which is crucial for the Leja-based methods. Let us clearly state that we use the state-of the-art KIOPS algorithm [15] for this work, which is different from the elementary Krylov-based method that we used in the previous study [11]. We describe the test problems in Sec.…”

Section: Introductionmentioning

confidence: 99%

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A comparison of Leja- and Krylov-based iterative schemes for Exponential Integrators

Deka¹,

Tokman²,

Einkemmer³

2022

Preprint

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Krylov-based algorithms have long been preferred to compute the matrix exponential and exponential-like functions appearing in exponential integrators. Of late, direct polynomial interpolation of the action of these exponentiallike functions have been shown to be competitive with the Krylov methods. We analyse the performance of the state-of-the-art Krylov algorithm, KIOPS, and the method of polynomial interpolation at Leja points for a number of exponential integrators for various test problems and with varying amounts of stiffness. Additionally, we investigate the performance of an iterative scheme that combines both the KIOPS and Leja approach, named LeKry, that shows substantial improvements over both the Leja-and Krylov-based methods for certain exponential integrators. Whilst we do manage to single out a favoured iterative scheme for each of the exponential integrators that we consider in this study, we do not find any conclusive evidence for preferring either KIOPS or Leja for different classes of exponential integrators. We are unable to identify a superior exponential integrator, one that performs better than all others, for most, if not all of the problems under consideration. We, however, do find that the performance significantly depends on the interplay between the iterative scheme and the specific exponential integrator under consideration.

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Section: Exprb43contrasting

confidence: 67%

Section: Epirk5p1mentioning

confidence: 77%