| Issue |
ESAIM: M2AN
Volume 56, Number 3, May-June 2022
|
|
|---|---|---|
| Page(s) | 743 - 766 | |
| DOI | https://doi.org/10.1051/m2an/2022018 | |
| Published online | 25 April 2022 | |
High order linearly implicit methods for evolution equations
1
Univ. Lille, Inria, CNRS, UMR 8524 – Laboratoire Paul Painlevé, F-59000 Lille, France
2
Université de Lorraine, CNRS, IECL, F-54000 Nancy, France
* Corresponding author: ingrid.lacroix@univ-lorraine.fr
Received:
28
October
2020
Accepted:
11
February
2022
This paper introduces a new class of numerical methods for the time integration of evolution equations set as Cauchy problems of ODEs or PDEs. The systematic design of these methods mixes the Runge–Kutta collocation formalism with collocation techniques, in such a way that the methods are linearly implicit and have high order. The fact that these methods are implicit allows to avoid CFL conditions when the large systems to integrate come from the space discretization of evolution PDEs. Moreover, these methods are expected to be efficient since they only require to solve one linear system of equations at each time step, and efficient techniques from the literature can be used to do so. After the introduction of the methods, we set suitable definitions of consistency and stability for these methods. This allows for a proof that arbitrarily high order linearly implicit methods exist and converge when applied to ODEs. Eventually, we perform numerical experiments on ODEs and PDEs that illustrate our theoretical results for ODEs, and compare our methods with standard methods for several evolution PDEs.
Mathematics Subject Classification: 65M12 / 65M70 / 65L20 / 65L06 / 81Q05 / 35Q41 / 35K05
Key words: Cauchy problems / evolution equations / time integration / numerical methods / high order / linearly implicit methods
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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