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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)
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author
organization
publishing date
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
2016-03-02 17:45:00
date last changed
2017-09-18 11:28:12
@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},
  keyword      = {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},
  volume       = {56},
  year         = {2016},
}