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Title: | Jacobian-free implicit MDRK methods for stiff systems of ODEs | Authors: | CHOUCHOULIS, Jeremy SCHUETZ, Jochen |
Issue Date: | 2024 | Publisher: | ELSEVIER | Source: | Applied Numerical Mathematics, 196 , p. 45 -61 | Abstract: | In this work, an approximate family of implicit multiderivative Runge-Kutta (MDRK) time integrators for stiff initial value problems is presented. The approximation procedure is based on the recent Approximate Implicit Taylor method (Baeza et al., 2020 [7]). As a Taylor method can be written in MDRK format, the novel family constitutes a multistage generalization. Two different alternatives are investigated for the computation of the higher order derivatives: either directly as part of the stage equation, or either as a separate formula for each derivative added on top of the stage equation itself. From linearizing through Newton's method, it turns out that the conditioning of the Newton matrix behaves significantly different for both cases. We show that direct computation results in a matrix with a conditioning that is highly dependent on the stiffness, increasing exponentially in the stiffness parameter with the amount of derivatives. Adding separate formulas has a more favorable behavior, the matrix conditioning being linearly dependent on the stiffness, regardless of the amount of derivatives. Despite increasing the Newton system significantly in size, through several numerical results it is demonstrated that doing so can be considerably beneficial. | Notes: | Chouchoulis, J (corresponding author), Hasselt Univ, Fac Sci, Agoralaan Gebouw D, B-3590 Diepenbeek, Belgium.; Chouchoulis, J (corresponding author), Hasselt Univ, Data Sci Inst, Agoralaan Gebouw D, B-3590 Diepenbeek, Belgium. jeremy.chouchoulis@uhasselt.be; jochen.schuetz@uhasselt.be |
Keywords: | Multiderivative Runge-Kutta;Jacobian-free;ODE integrator | Document URI: | http://hdl.handle.net/1942/41740 | ISSN: | 0168-9274 | e-ISSN: | 1873-5460 | DOI: | 10.1016/j.apnum.2023.10.007 | ISI #: | 001108088900001 | Rights: | 2023 IMACS. Published by Elsevier B.V. All rights reserved. | Category: | A1 | Type: | Journal Contribution |
Appears in Collections: | Research publications |
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1-s2.0-S0168927423002672-main (1).pdf Restricted Access | Published version | 1.51 MB | Adobe PDF | View/Open Request a copy |
ApproxMDRK.pdf | Peer-reviewed author version | 650.27 kB | Adobe PDF | View/Open |
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