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Resolving Entropy Growth from Iterative Methods

Linders, Viktor LU ; Ranocha, Hendrik LU and Birken, Philipp LU (2023)
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)
Please use this url to cite or link to this publication:
author
; and
organization
publishing date
type
Working paper/Preprint
publication status
published
subject
publisher
arXiv.org
DOI
10.48550/arXiv.2302.13579
project
Olinjära vågor och entropistabil iteration
language
English
LU publication?
yes
id
7a549c89-9e4b-48dc-ad1a-c1b437e49966
date added to LUP
2023-09-01 11:31:41
date last changed
2023-10-11 09:10:38
@misc{7a549c89-9e4b-48dc-ad1a-c1b437e49966,
  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 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.}},
  author       = {{Linders, Viktor and Ranocha, Hendrik and Birken, Philipp}},
  language     = {{eng}},
  note         = {{Preprint}},
  publisher    = {{arXiv.org}},
  title        = {{Resolving Entropy Growth from Iterative Methods}},
  url          = {{http://dx.doi.org/10.48550/arXiv.2302.13579}},
  doi          = {{10.48550/arXiv.2302.13579}},
  year         = {{2023}},
}