Issues in measurepreserving three dimensional flow integrators: Selfadjointness, reversibility, and nonuniform time stepping
Abstract
Properties of integration schemes for solenoidal fields in three dimensions are studied, with a focus on integrating magnetic field lines in a plasma using adaptive time stepping. It is shown that implicit midpoint (IM) and a scheme we call threedimensional leapfrog (LF) can do a good job (in the sense of preserving KAM tori) of integrating fields that are reversible, or (for LF) have a 'special divergencefree' property. We review the notion of a selfadjoint scheme, showing that such schemes are at least second order accurate and can always be formed by composing an arbitrary scheme with its adjoint. We also review the concept of reversibility, showing that a reversible but not exactly volumepreserving scheme can lead to a fractal invariant measure in a chaotic region, although this property may not often be observable. We also show numerical results indicating that the IM and LF schemes can fail to preserve KAM tori when the reversibility property (and the SDF property for LF) of the field is broken. We discuss extensions to measure preserving flows, the integration of magnetic field lines in a plasma and the integration of rays for several plasma waves. The main new result of this paper relatesmore »
 Authors:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1193292
 Report Number(s):
 LAUR1428212
Journal ID: ISSN 1070664X; PHPAEN
 Grant/Contract Number:
 AC5206NA25396
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Physics of Plasmas
 Additional Journal Information:
 Journal Volume: 22; Journal Issue: 3; Journal ID: ISSN 1070664X
 Publisher:
 American Institute of Physics (AIP)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; volume preserving integrator; selfadjoint integrator; reversible integrator; adaptive time stepping
Citation Formats
Finn, John M. Issues in measurepreserving three dimensional flow integrators: Selfadjointness, reversibility, and nonuniform time stepping. United States: N. p., 2015.
Web. doi:10.1063/1.4914839.
Finn, John M. Issues in measurepreserving three dimensional flow integrators: Selfadjointness, reversibility, and nonuniform time stepping. United States. doi:10.1063/1.4914839.
Finn, John M. 2015.
"Issues in measurepreserving three dimensional flow integrators: Selfadjointness, reversibility, and nonuniform time stepping". United States.
doi:10.1063/1.4914839. https://www.osti.gov/servlets/purl/1193292.
@article{osti_1193292,
title = {Issues in measurepreserving three dimensional flow integrators: Selfadjointness, reversibility, and nonuniform time stepping},
author = {Finn, John M.},
abstractNote = {Properties of integration schemes for solenoidal fields in three dimensions are studied, with a focus on integrating magnetic field lines in a plasma using adaptive time stepping. It is shown that implicit midpoint (IM) and a scheme we call threedimensional leapfrog (LF) can do a good job (in the sense of preserving KAM tori) of integrating fields that are reversible, or (for LF) have a 'special divergencefree' property. We review the notion of a selfadjoint scheme, showing that such schemes are at least second order accurate and can always be formed by composing an arbitrary scheme with its adjoint. We also review the concept of reversibility, showing that a reversible but not exactly volumepreserving scheme can lead to a fractal invariant measure in a chaotic region, although this property may not often be observable. We also show numerical results indicating that the IM and LF schemes can fail to preserve KAM tori when the reversibility property (and the SDF property for LF) of the field is broken. We discuss extensions to measure preserving flows, the integration of magnetic field lines in a plasma and the integration of rays for several plasma waves. The main new result of this paper relates to nonuniform time stepping for volumepreserving flows. We investigate two potential schemes, both based on the general method of Ref. [11], in which the flow is integrated in split time steps, each Hamiltonian in two dimensions. The first scheme is an extension of the method of extended phase space, a wellproven method of symplectic integration with nonuniform time steps. This method is found not to work, and an explanation is given. The second method investigated is a method based on transformation to canonical variables for the two splitstep Hamiltonian systems. This method, which is related to the method of noncanonical generating functions of Ref. [35], appears to work very well.},
doi = {10.1063/1.4914839},
journal = {Physics of Plasmas},
number = 3,
volume = 22,
place = {United States},
year = 2015,
month = 3
}
Web of Science

Properties of integration schemes for solenoidal fields in three dimensions are studied, with a focus on integrating magnetic field lines in a plasma using adaptive time stepping. It is shown that implicit midpoint (IM) and a scheme we call threedimensional leapfrog (LF) can do a good job (in the sense of preserving KAM tori) of integrating fields that are reversible, or (for LF) have a “special divergencefree” (SDF) property. We review the notion of a selfadjoint scheme, showing that such schemes are at least second order accurate and can always be formed by composing an arbitrary scheme with its adjoint.more »

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