High-order splitting schemes for semilinear evolution equations
Hansen, Eskil LU
and Ostermann, Alexander
(2016)
In BIT Numerical Mathematics
56(4).
p.1303-1316
- Abstract
- We first derive necessary and sufficient stiff order conditions, up to order four, for exponential splitting schemes applied to semilinear evolution equations. The main idea is to identify the local splitting error as a sum of quadrature errors. The order conditions of the quadrature rules then yield the stiff order conditions in an explicit fashion, similarly to that of Runge–Kutta schemes. Furthermore, the derived stiff conditions coincide with the classical non-stiff conditions. Secondly, we propose an abstract convergence analysis, where the linear part of the vector field is assumed to generate a group or a semigroup and the nonlinear part is assumed to be smooth and to satisfy a set of compatibility requirements. Concrete... (More)
- We first derive necessary and sufficient stiff order conditions, up to order four, for exponential splitting schemes applied to semilinear evolution equations. The main idea is to identify the local splitting error as a sum of quadrature errors. The order conditions of the quadrature rules then yield the stiff order conditions in an explicit fashion, similarly to that of Runge–Kutta schemes. Furthermore, the derived stiff conditions coincide with the classical non-stiff conditions. Secondly, we propose an abstract convergence analysis, where the linear part of the vector field is assumed to generate a group or a semigroup and the nonlinear part is assumed to be smooth and to satisfy a set of compatibility requirements. Concrete applications include nonlinear wave equations and diffusion-reaction processes. The convergence analysis also extends to the case where the nonlinear flows in the exponential splitting scheme are approximated by a sufficiently accurate one-step method. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/8598694
- author
- Hansen, Eskil
LU
and Ostermann, Alexander
- organization
- publishing date
- 2016
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Splitting schemes, Exponential splitting, Semilinear evolution equations, High-order methods, Stiff orders, Convergence
- in
- BIT Numerical Mathematics
- volume
- 56
- issue
- 4
- pages
- 14 pages
- publisher
- Springer
- external identifiers
-
- scopus:84997816038
- wos:000388968500008
- ISSN
- 0006-3835
- DOI
- 10.1007/s10543-016-0604-2
- language
- English
- LU publication?
- yes
- id
- d45f8dbc-5a34-493f-9343-51b9a446fcdb (old id 8598694)
- alternative location
- http://www.maths.lth.se/na/staff/eskil/dataEskil/articles/Highordersplit.pdf
- date added to LUP
- 2016-04-01 09:52:11
- date last changed
- 2024-10-06 14:36:37
@article{d45f8dbc-5a34-493f-9343-51b9a446fcdb,
abstract = {{We first derive necessary and sufficient stiff order conditions, up to order four, for exponential splitting schemes applied to semilinear evolution equations. The main idea is to identify the local splitting error as a sum of quadrature errors. The order conditions of the quadrature rules then yield the stiff order conditions in an explicit fashion, similarly to that of Runge–Kutta schemes. Furthermore, the derived stiff conditions coincide with the classical non-stiff conditions. Secondly, we propose an abstract convergence analysis, where the linear part of the vector field is assumed to generate a group or a semigroup and the nonlinear part is assumed to be smooth and to satisfy a set of compatibility requirements. Concrete applications include nonlinear wave equations and diffusion-reaction processes. The convergence analysis also extends to the case where the nonlinear flows in the exponential splitting scheme are approximated by a sufficiently accurate one-step method.}},
author = {{Hansen, Eskil and Ostermann, Alexander}},
issn = {{0006-3835}},
keywords = {{Splitting schemes; Exponential splitting; Semilinear evolution equations; High-order methods; Stiff orders; Convergence}},
language = {{eng}},
number = {{4}},
pages = {{1303--1316}},
publisher = {{Springer}},
series = {{BIT Numerical Mathematics}},
title = {{High-order splitting schemes for semilinear evolution equations}},
url = {{http://dx.doi.org/10.1007/s10543-016-0604-2}},
doi = {{10.1007/s10543-016-0604-2}},
volume = {{56}},
year = {{2016}},
}