Resolving entropy growth from iterative methods
(2023) In BIT Numerical Mathematics 63(4).- Abstract
We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton’s method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider... (More)
We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton’s method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers’ equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.
(Less)
- author
- Linders, Viktor LU ; Ranocha, Hendrik LU and Birken, Philipp LU
- organization
- publishing date
- 2023-08-03
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Dispersive wave equations, Entropy conservation, Implicit methods, Iterative methods
- in
- BIT Numerical Mathematics
- volume
- 63
- issue
- 4
- article number
- 45
- pages
- 26 pages
- publisher
- Springer
- external identifiers
-
- scopus:85171363710
- ISSN
- 0006-3835
- DOI
- 10.1007/s10543-023-00992-w
- project
- Olinjära vågor och entropistabil iteration
- language
- English
- LU publication?
- yes
- additional info
- Publisher Copyright: © 2023, The Author(s).
- id
- 1880086a-f5e0-4cc9-902f-e8047feeab32
- date added to LUP
- 2023-10-02 10:33:26
- date last changed
- 2023-10-09 16:44:05
@article{1880086a-f5e0-4cc9-902f-e8047feeab32,
abstract = {{<p>We consider entropy conservative and dissipative discretizations of nonlinear conservation laws with implicit time discretizations and investigate the influence of iterative methods used to solve the arising nonlinear equations. We show that Newton’s method can turn an entropy dissipative scheme into an anti-dissipative one, even when the iteration error is smaller than the time integration error. We explore several remedies, of which the most performant is a relaxation technique, originally designed to fix entropy errors in time integration methods. Thus, relaxation works well in consort with iterative solvers, provided that the iteration errors are on the order of the time integration method. To corroborate our findings, we consider Burgers’ equation and nonlinear dispersive wave equations. We find that entropy conservation results in more accurate numerical solutions than non-conservative schemes, even when the tolerance is an order of magnitude larger.</p>}},
author = {{Linders, Viktor and Ranocha, Hendrik and Birken, Philipp}},
issn = {{0006-3835}},
keywords = {{Dispersive wave equations; Entropy conservation; Implicit methods; Iterative methods}},
language = {{eng}},
month = {{08}},
number = {{4}},
publisher = {{Springer}},
series = {{BIT Numerical Mathematics}},
title = {{Resolving entropy growth from iterative methods}},
url = {{http://dx.doi.org/10.1007/s10543-023-00992-w}},
doi = {{10.1007/s10543-023-00992-w}},
volume = {{63}},
year = {{2023}},
}