# Issues in measure-preserving three dimensional flow integrators: Self-adjointness, reversibility, and non-uniform 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 three-dimensional 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 divergence-free' property. We review the notion of a self-adjoint 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 volume-preserving 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):
- LA-UR-14-28212

Journal ID: ISSN 1070-664X; PHPAEN

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physics of Plasmas

- Additional Journal Information:
- Journal Volume: 22; Journal Issue: 3; Journal ID: ISSN 1070-664X

- Publisher:
- American Institute of Physics (AIP)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; volume preserving integrator; self-adjoint integrator; reversible integrator; adaptive time stepping

### Citation Formats

```
Finn, John M.
```*Issues in measure-preserving three dimensional flow integrators: Self-adjointness, reversibility, and non-uniform time stepping*. United States: N. p., 2015.
Web. doi:10.1063/1.4914839.

```
Finn, John M.
```*Issues in measure-preserving three dimensional flow integrators: Self-adjointness, reversibility, and non-uniform time stepping*. United States. doi:10.1063/1.4914839.

```
Finn, John M. Sun .
"Issues in measure-preserving three dimensional flow integrators: Self-adjointness, reversibility, and non-uniform time stepping". United States.
doi:10.1063/1.4914839. https://www.osti.gov/servlets/purl/1193292.
```

```
@article{osti_1193292,
```

title = {Issues in measure-preserving three dimensional flow integrators: Self-adjointness, reversibility, and non-uniform 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 three-dimensional 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 divergence-free' property. We review the notion of a self-adjoint 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 volume-preserving 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 non-uniform time stepping for volume-preserving 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 well-proven method of symplectic integration with non-uniform 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 split-step Hamiltonian systems. This method, which is related to the method of non-canonical 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 = {Sun Mar 01 00:00:00 EST 2015},

month = {Sun Mar 01 00:00:00 EST 2015}

}

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