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 47 articles , 1 standard article )

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  1. Altmann, Robert; Zimmer, Christoph: Exponential integrators for semi-linear parabolic problems with linear constraints (2020)
  2. Bertoli, Guillaume; Vilmart, Gilles: Strang splitting method for semilinear parabolic problems with inhomogeneous boundary conditions: a correction based on the flow of the nonlinearity (2020)
  3. Buvoli, Tommaso: A class of exponential integrators based on spectral deferred correction (2020)
  4. Caliari, Marco; Cassini, Fabio; Zivcovich, Franco: Approximation of the matrix exponential for matrices with a skinny field of values (2020)
  5. Hoang, Thi-Thao-Phuong; Ju, Lili; Wang, Zhu: Nonoverlapping localized exponential time differencing methods for diffusion problems (2020)
  6. Jawecki, Tobias; Auzinger, Winfried; Koch, Othmar: Computable upper error bounds for Krylov approximations to matrix exponentials and associated (\varphi)-functions (2020)
  7. Luan, Vu Thai; Chinomona, Rujeko; Reynolds, Daniel R.: A new class of high-order methods for multirate differential equations (2020)
  8. Narayanamurthi, Mahesh; Sandu, Adrian: Efficient implementation of partitioned stiff exponential Runge-Kutta methods (2020)
  9. Isherwood, Leah; Grant, Zachary J.; Gottlieb, Sigal: Strong stability preserving integrating factor two-step Runge-Kutta methods (2019)
  10. Luan, Vu Thai; Pudykiewicz, Janusz A.; Reynolds, Daniel R.: Further development of efficient and accurate time integration schemes for meteorological models (2019)
  11. Meltzer, A. Y.: An accurate approximation of exponential integrators for the Schrödinger equation (2019)
  12. Narayanamurthi, Mahesh; Tranquilli, Paul; Sandu, Adrian; Tokman, Mayya: EPIRK-(W) and EPIRK-(K) time discretization methods (2019)
  13. Cano, Begoña; Moreta, María Jesús: Exponential quadrature rules without order reduction for integrating linear initial boundary value problems (2018)
  14. 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)
  15. Gaudreault, Stéphane; Rainwater, Greg; Tokman, Mayya: KIOPS: a fast adaptive Krylov subspace solver for exponential integrators (2018)
  16. Isherwood, Leah; Grant, Zachary J.; Gottlieb, Sigal: Strong stability preserving integrating factor Runge-Kutta methods (2018)
  17. Rostami, Minghao W.; Xue, Fei: Robust linear stability analysis and a new method for computing the action of the matrix exponential (2018)
  18. Wu, Gang; Pang, Hong-Kui; Sun, Jiang-Li: A shifted block FOM algorithm with deflated restarting for matrix exponential computations (2018)
  19. Einkemmer, Lukas; Tokman, Mayya; Loffeld, John: On the performance of exponential integrators for problems in magnetohydrodynamics (2017)
  20. Li, Dongping; Cong, Yuhao: Approximation of the linear combination of (\varphi)-functions using the block shift-and-invert Krylov subspace method (2017)

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