LeXInt: Package for Exponential Integrators Employing Leja Interpolation

Deka, Pranab J.; Einkemmer, Lukas; Tokman, Mayya

doi:10.2139/ssrn.4243380

Cited by 2 publications

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“…The number of matrix-vector products computed is proportional to the number of Leja points needed for the polynomial to converge and can thus be used as a proxy of the computational cost. We use the (publicly available) LeXInt library [16] to compute the exponential of the matrix applied to a vector using the Leja interpolation scheme.…”

Section: Computation Of the Matrix Exponential Using Leja Interpolationmentioning

confidence: 99%

Exponential methods for anisotropic diffusion

Deka¹,

Einkemmer²,

Kissmann³

2022

Preprint

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The anisotropic diffusion equation is of crucial importance in understanding cosmic ray diffusion across the Galaxy and its interplay with the Galactic magnetic field. This diffusion term contributes to the highly stiff nature of the CR transport equation. In order to conduct numerical simulations of time-dependent cosmic ray transport, implicit integrators have been traditionally favoured over the CFL-bound explicit integrators in order to be able to take large step sizes. We propose exponential methods that directly compute the exponential of the matrix to solve the linear anisotropic diffusion equation. These methods allow us to take even larger step sizes; in certain cases, we are able to choose a step size as large as the simulation time, i.e., only one time step. This can substantially speed-up the simulations whilst generating highly accurate solutions (l2 error ≤ 10 −10 ). Additionally, we test an approach based on extracting a constant coefficient from the anisotropic diffusion equation where the constant coefficient term is solved implicitly or exponentially and the remainder is treated using some explicit method. We find that this approach, for linear problems, is unable to improve on the exponential-based methods that directly evaluate the matrix exponential.

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Section: Computation Of the Matrix Exponential Using Leja Interpolationmentioning

confidence: 99%

Exponential methods for anisotropic diffusion

Deka¹,

Einkemmer²,

Kissmann³

2022

Preprint

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“…The Python version of the Leja method for exponential integrators is a publicly available software. Further details on the algorithm and the code can be found in Deka et al [13]. Computing the exponential an augmented matrix clearly results in performance loss.…”

Section: Softwarementioning

confidence: 99%

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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LeXInt: Package for Exponential Integrators Employing Leja Interpolation

Cited by 2 publications

References 0 publications

Exponential methods for anisotropic diffusion

Exponential methods for anisotropic diffusion

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

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