Algorithm 919

Algorithm 919: A Krylov Subspace Algorithm for Evaluating the ϕ-Functions Appearing in Exponential Integrators. We develop an algorithm for computing the solution of a large system of linear ordinary differential equations (ODEs) with polynomial inhomogeneity. This is equivalent to computing the action of a certain matrix function on the vector representing the initial condition. The matrix function is a linear combination of the matrix exponential and other functions related to the exponential (the so-called ϕ-functions). Such computations are the major computational burden in the implementation of exponential integrators, which can solve general ODEs. Our approach is to compute the action of the matrix function by constructing a Krylov subspace using Arnoldi or Lanczos iteration and projecting the function on this subspace. This is combined with time-stepping to prevent the Krylov subspace from growing too large. The algorithm is fully adaptive: it varies both the size of the time steps and the dimension of the Krylov subspace to reach the required accuracy. We implement this algorithm in the matlab function phipm and we give instructions on how to obtain and use this function. Various numerical experiments show that the phipm function is often significantly more efficient than the state-of-the-art.

This software is also peer reviewed by journal TOMS.

References in zbMATH (referenced in 34 articles , 1 standard article )

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  1. Narayanamurthi, Mahesh; Tranquilli, Paul; Sandu, Adrian; Tokman, Mayya: EPIRK-(W) and EPIRK-(K) time discretization methods (2019)
  2. Cano, Begoña; Moreta, María Jesús: Exponential quadrature rules without order reduction for integrating linear initial boundary value problems (2018)
  3. Galanin, M. P.; Konev, S. A.: Development and application of an exponential method for integrating stiff systems based on the classical Runge-Kutta method (2018)
  4. Gaudreault, Stéphane; Rainwater, Greg; Tokman, Mayya: KIOPS: a fast adaptive Krylov subspace solver for exponential integrators (2018)
  5. Isherwood, Leah; Grant, Zachary J.; Gottlieb, Sigal: Strong stability preserving integrating factor Runge-Kutta methods (2018)
  6. Rostami, Minghao W.; Xue, Fei: Robust linear stability analysis and a new method for computing the action of the matrix exponential (2018)
  7. Wu, Gang; Pang, Hong-Kui; Sun, Jiang-Li: A shifted block FOM algorithm with deflated restarting for matrix exponential computations (2018)
  8. Einkemmer, Lukas; Tokman, Mayya; Loffeld, John: On the performance of exponential integrators for problems in magnetohydrodynamics (2017)
  9. Li, Yiqun; Wu, Boying; Leok, Melvin: Spectral variational integrators for semi-discrete Hamiltonian wave equations (2017)
  10. Lord, G. J.; Stone, D.: New efficient substepping methods for exponential timestepping (2017)
  11. Luan, Vu Thai: Fourth-order two-stage explicit exponential integrators for time-dependent PDEs (2017)
  12. Vo, H. D.; Sidje, R. B.: Implementation of variable parameters in the Krylov-based finite state projection for solving the chemical master equation (2017)
  13. Bader, Philipp; Blanes, Sergio; Casas, Fernando; Ponsoda, Enrique: Efficient numerical integration of (N)th-order non-autonomous linear differential equations (2016)
  14. Botchev, Mikhail A.: Krylov subspace exponential time domain solution of Maxwell’s equations in photonic crystal modeling (2016)
  15. Caliari, Marco; Kandolf, Peter; Ostermann, Alexander; Rainer, Stefan: The Leja method revisited: backward error analysis for the matrix exponential (2016)
  16. Cong, Yuhao; Li, Dongping: Block Krylov subspace methods for approximating the linear combination of (\varphi)-functions arising in exponential integrators (2016)
  17. Farquhar, Megan E.; Moroney, Timothy J.; Yang, Qianqian; Turner, Ian W.: GPU accelerated algorithms for computing matrix function vector products with applications to exponential integrators and fractional diffusion (2016)
  18. Gaudreault, Stéphane; Pudykiewicz, Janusz A.: An efficient exponential time integration method for the numerical solution of the shallow water equations on the sphere (2016)
  19. Li, Dongping; Cong, Yuhao; Xia, Kaifeng: Flexible exponential integration methods for large systems of differential equations (2016)
  20. Wu, Gang; Zhang, Lu; Xu, Ting-ting: A framework of the harmonic Arnoldi method for evaluating (\varphi)-functions with applications to exponential integrators (2016)

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