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Locally conservative and flux consistent iterative methods

Linders, Viktor LU and Birken, Philipp LU (2023)
Abstract
Conservation and consistency are fundamental properties of discretizations of systems of hyperbolic conservation laws. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux consistent iterations. These concepts are of both theoretical and practical importance: Based on recent work by the authors, it is shown that pseudo-time iterations using explicit Runge-Kutta methods are locally conservative but not necessarily flux consistent. An extension of the Lax-Wendroff theorem is presented, revealing convergence towards weak solutions of a temporally retarded system of conservation laws. Each equation is modified in the same way, namely by a particular scalar factor multiplying the... (More)
Conservation and consistency are fundamental properties of discretizations of systems of hyperbolic conservation laws. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux consistent iterations. These concepts are of both theoretical and practical importance: Based on recent work by the authors, it is shown that pseudo-time iterations using explicit Runge-Kutta methods are locally conservative but not necessarily flux consistent. An extension of the Lax-Wendroff theorem is presented, revealing convergence towards weak solutions of a temporally retarded system of conservation laws. Each equation is modified in the same way, namely by a particular scalar factor multiplying the spatial flux terms. A technique for enforcing flux consistency, and thereby recovering convergence, is presented. Further, local conservation is established for all Krylov subspace methods, with and without restarts, and for Newton's method under certain assumptions on the discretization. Thus it is shown that Newton-Krylov methods are locally conservative, although not necessarily flux consistent. Numerical experiments with the 2D compressible Euler equations corroborate the theoretical results. Further numerical investigations of the impact of flux consistency on Newton-Krylov methods indicate that its effect is case dependent, and diminishes as the number of iterations grow. (Less)
Please use this url to cite or link to this publication:
author
and
organization
publishing date
type
Working paper/Preprint
publication status
in press
subject
keywords
Iterative methods, Conservation laws, Lax-Wendroff theorem, Pseudo-time iterations, Newton-Krylov methods
pages
28 pages
DOI
10.48550/arXiv.2206.10943
language
English
LU publication?
yes
id
eaa82e92-aee8-4fcd-9c4c-4fd9ba068c36
date added to LUP
2023-09-01 11:36:15
date last changed
2023-10-18 10:53:14
@misc{eaa82e92-aee8-4fcd-9c4c-4fd9ba068c36,
  abstract     = {{Conservation and consistency are fundamental properties of discretizations of systems of hyperbolic conservation laws. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux consistent iterations. These concepts are of both theoretical and practical importance: Based on recent work by the authors, it is shown that pseudo-time iterations using explicit Runge-Kutta methods are locally conservative but not necessarily flux consistent. An extension of the Lax-Wendroff theorem is presented, revealing convergence towards weak solutions of a temporally retarded system of conservation laws. Each equation is modified in the same way, namely by a particular scalar factor multiplying the spatial flux terms. A technique for enforcing flux consistency, and thereby recovering convergence, is presented. Further, local conservation is established for all Krylov subspace methods, with and without restarts, and for Newton's method under certain assumptions on the discretization. Thus it is shown that Newton-Krylov methods are locally conservative, although not necessarily flux consistent. Numerical experiments with the 2D compressible Euler equations corroborate the theoretical results. Further numerical investigations of the impact of flux consistency on Newton-Krylov methods indicate that its effect is case dependent, and diminishes as the number of iterations grow.}},
  author       = {{Linders, Viktor and Birken, Philipp}},
  keywords     = {{Iterative methods; Conservation laws; Lax-Wendroff theorem; Pseudo-time iterations; Newton-Krylov methods}},
  language     = {{eng}},
  note         = {{Preprint}},
  title        = {{Locally conservative and flux consistent iterative methods}},
  url          = {{http://dx.doi.org/10.48550/arXiv.2206.10943}},
  doi          = {{10.48550/arXiv.2206.10943}},
  year         = {{2023}},
}