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 66 articles , 1 standard article )

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  1. Brachet, M.; Croisille, J.-P.: A center compact scheme for the shallow water equations on the sphere (2022)
  2. Buvoli, Tommaso; Minion, Michael L.: On the stability of exponential integrators for non-diffusive equations (2022)
  3. Jawecki, Tobias: A study of defect-based error estimates for the Krylov approximation of (\varphi)-functions (2022)
  4. Li, Dongping; Yang, Siyu; Lan, Jiamei: Efficient and accurate computation for the (\varphi)-functions arising from exponential integrators (2022)
  5. Bertoli, Guillaume; Besse, Christophe; Vilmart, Gilles: Superconvergence of the Strang splitting when using the Crank-Nicolson scheme for parabolic PDEs with Dirichlet and oblique boundary conditions (2021)
  6. Botchev, Mike A.; Knizhnerman, Leonid; Tyrtyshnikov, Eugene E.: Residual and restarting in Krylov subspace evaluation of the (\varphi) function (2021)
  7. Buvoli, Tommaso: Exponential polynomial block methods (2021)
  8. Chen, Hao; Sun, Hai-Wei: A dimensional splitting exponential time differencing scheme for multidimensional fractional Allen-Cahn equations (2021)
  9. Du, Qiang; Ju, Lili; Li, Xiao; Qiao, Zhonghua: Maximum bound principles for a class of semilinear parabolic equations and exponential time-differencing schemes (2021)
  10. Kang, Shinhoo; Bui-Thanh, Tan: A scalable exponential-DG approach for nonlinear conservation laws: with application to Burger and Euler equations (2021)
  11. Luan, Vu Thai: Efficient exponential Runge-Kutta methods of high order: construction and implementation (2021)
  12. Luan, Vu Thai; Michels, Dominik L.: Efficient exponential time integration for simulating nonlinear coupled oscillators (2021)
  13. Meng, Xucheng; Hoang, Thi-Thao-phuong; Wang, Zhu; Ju, Lili: Localized exponential time differencing method for shallow water equations: algorithms and numerical study (2021)
  14. Muñoz-Matute, Judit; Pardo, David; Demkowicz, Leszek: A DPG-based time-marching scheme for linear hyperbolic problems (2021)
  15. Muñoz-Matute, Judit; Pardo, David; Demkowicz, Leszek: Equivalence between the DPG method and the exponential integrators for linear parabolic problems (2021)
  16. Naranjo-Noda, F. S.; Jimenez, J. C.: Locally linearized Runge-Kutta method of Dormand and Prince for large systems of initial value problems (2021)
  17. Narayanamurthi, Mahesh; Sandu, Adrian: Partitioned exponential methods for coupled multiphysics systems (2021)
  18. Altmann, Robert; Zimmer, Christoph: Exponential integrators for semi-linear parabolic problems with linear constraints (2020)
  19. 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)
  20. Buvoli, Tommaso: A class of exponential integrators based on spectral deferred correction (2020)

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