A new class of exponential propagation iterative methods of Runge–Kutta type (EPIRK)

Tokman, Mayya

doi:10.1016/j.jcp.2011.08.023

Cited by 53 publications

(80 citation statements)

References 30 publications

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“…Constant coefficients α ij , β ij and p ijk can be derived using either classical or stiff order conditions and methods of up to order five have been proposed in [44,33]. Note that since we are interested in problems where J n ∈ R N×N with N >> 1, the largest per-time step computational cost of (2) lies in evaluating the matrix function-vector products…”

Section: Approximating ϕ-Functions Within Exponential Integratorsmentioning

confidence: 99%

KIOPS: A fast adaptive Krylov subspace solver for exponential integrators

Gaudreault¹,

Rainwater²,

Tokman³

2018

Journal of Computational Physics

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Matrix exponentialThis paper presents a new algorithm KIOPS for computing linear combinations of ϕ-functions that appear in exponential integrators. This algorithm is suitable for largescale problems in computational physics where little or no information about the spectrum or norm of the Jacobian matrix is known a priori. We first show that such problems can be solved efficiently by computing a single exponential of a modified matrix. Then our approach is to compute an appropriate basis for the Krylov subspace using the incomplete orthogonalization procedure and project the matrix exponential on this subspace. We also present a novel adaptive procedure that significantly reduces the computational complexity of exponential integrators. Our numerical experiments demonstrate that KIOPS outperforms the current state-of-the-art adaptive Krylov algorithm phipm. Crown

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Section: Approximating ϕ-Functions Within Exponential Integratorsmentioning

confidence: 99%

KIOPS: A fast adaptive Krylov subspace solver for exponential integrators

Gaudreault¹,

Rainwater²,

Tokman³

2018

Journal of Computational Physics

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“…Exponential integrators (EI) have been studied for several decades [2,3]: for generic introductions into EIs and for various formulations of EIs, see [4,5,6,7,8]. Most relevant to the present paper, Clancy et al [9] applied EI in the context of Laplace transforms to gain higher accuracy.…”

Section: Exponential Integratorsmentioning

confidence: 99%

Exponential integrators with parallel-in-time rational approximations for the shallow-water equations on the rotating sphere

2019

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High-performance computing trends towards many-core systems are expected to continue over the next decade. As a result, parallel-in-time methods, mathematical formulations which exploit additional degrees of parallelism in the time dimension, have gained increasing interest in recent years. In this work we study a massively parallel rational approximation of exponential integrators (REXI). This method replaces a time integration of stiff linear oscillatory and diffusive systems by the sum of the solutions of many decoupled systems, which can be solved in parallel. Previous numerical studies showed that this reformulation allows taking arbitrarily long time steps for the linear oscillatory parts.The present work studies the non-linear shallow-water equations on the rotating sphere, a simplified system of equations used to study properties of space and time discretization methods in the context of atmospheric simulations. After introducing time integrators, we first compare the time step sizes to the errors in the simulation, discussing pros and cons of different formulations of REXI. Here, REXI already shows superior properties compared to explicit and implicit time stepping methods. Additionally, we present wallclock-time-to-error results revealing the sweet spots of REXI obtaining either an over 6× higher accuracy within the same time frame or an about 3× reduced time-to-solution for a similar error threshold. Our results motivate further explorations of REXI for operational weather/climate systems.

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“…It surfaced in computational electrodynamics in the 1990s [42] and was rediscovered for numerical solutions of nonlinear differential equations later [37,43,[71][72][73].…”

Section: Exponential Time Differencingmentioning

confidence: 99%

Exponential integrators in time-dependent density-functional calculations

2017

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The integrating factor and exponential time differencing methods are implemented and tested for solving the time-dependent Kohn-Sham equations. Popular time propagation methods used in physics, as well as other robust numerical approaches, are compared to these exponential integrator methods in order to judge the relative merit of the computational schemes. We determine an improvement in accuracy of multiple orders of magnitude when describing dynamics driven primarily by a nonlinear potential. For cases of dynamics driven by a time-dependent external potential, the accuracy of the exponential integrator methods are less enhanced but still match or outperform the best of the conventional methods tested.

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A new class of exponential propagation iterative methods of Runge–Kutta type (EPIRK)

Cited by 53 publications

References 30 publications

KIOPS: A fast adaptive Krylov subspace solver for exponential integrators

KIOPS: A fast adaptive Krylov subspace solver for exponential integrators

Exponential integrators with parallel-in-time rational approximations for the shallow-water equations on the rotating sphere

Exponential integrators in time-dependent density-functional calculations

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