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## Multigrid treatment of implicit continuum diffusion

## Abstract

Implicit treatment of diffusive terms of various differential orders common in continuum mechanics modeling, such as computational fluid dynamics, is investigated with spectral and multigrid algorithms in non-periodic 2D domains. In doubly periodic time dependent problems these terms are handled efficiently by spectral methods, but in non-periodic systems solved with distributed memory parallel computing and 2D domain decomposition, this efficiency is lost for a large number of processors. Here, we built and present here a multigrid algorithm for these types of problems that outperforms a spectral solution employing the highly optimized FFTW library. This solver is suitable for high performance computing and may be able to efficiently treat implicit diffusion of arbitrary order by introducing auxiliary equations of lower order. We test these solvers for fourth and sixth order diffusion with harmonic test functions as well as turbulent 2D magnetohydrodynamic simulations. It is also shown that an anisotropic operator without mixed-derivative terms improves model accuracy and speed, and we examine the impact that the various diffusion operators have on the energy, the enstrophy, and the qualitative aspect of a simulation.

- Authors:

- Dartmouth College, Hanover, NH (United States). Dept. of Physics and Astronomy
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Publication Date:

- Research Org.:
- Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1497964

- Report Number(s):
- LLNL-JRNL-756939

Journal ID: ISSN 0010-4655; 944576

- Grant/Contract Number:
- AC52-07NA27344; SC0010508

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Computer Physics Communications

- Additional Journal Information:
- Journal Volume: 236; Journal Issue: C; Journal ID: ISSN 0010-4655

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 70 PLASMA PHYSICS AND FUSION TECHNOLOGY; Hyperdiffusion; Hyperviscosity; Laplace–Beltrami; bi-Helmholtz; tri-Helmholtz; Multigrid

### Citation Formats

```
Francisquez, M., Zhu, B., and Rogers, B. N. Multigrid treatment of implicit continuum diffusion. United States: N. p., 2019.
Web. doi:10.1016/j.cpc.2018.10.022.
```

```
Francisquez, M., Zhu, B., & Rogers, B. N. Multigrid treatment of implicit continuum diffusion. United States. doi:10.1016/j.cpc.2018.10.022.
```

```
Francisquez, M., Zhu, B., and Rogers, B. N. Fri .
"Multigrid treatment of implicit continuum diffusion". United States. doi:10.1016/j.cpc.2018.10.022.
```

```
@article{osti_1497964,
```

title = {Multigrid treatment of implicit continuum diffusion},

author = {Francisquez, M. and Zhu, B. and Rogers, B. N.},

abstractNote = {Implicit treatment of diffusive terms of various differential orders common in continuum mechanics modeling, such as computational fluid dynamics, is investigated with spectral and multigrid algorithms in non-periodic 2D domains. In doubly periodic time dependent problems these terms are handled efficiently by spectral methods, but in non-periodic systems solved with distributed memory parallel computing and 2D domain decomposition, this efficiency is lost for a large number of processors. Here, we built and present here a multigrid algorithm for these types of problems that outperforms a spectral solution employing the highly optimized FFTW library. This solver is suitable for high performance computing and may be able to efficiently treat implicit diffusion of arbitrary order by introducing auxiliary equations of lower order. We test these solvers for fourth and sixth order diffusion with harmonic test functions as well as turbulent 2D magnetohydrodynamic simulations. It is also shown that an anisotropic operator without mixed-derivative terms improves model accuracy and speed, and we examine the impact that the various diffusion operators have on the energy, the enstrophy, and the qualitative aspect of a simulation.},

doi = {10.1016/j.cpc.2018.10.022},

journal = {Computer Physics Communications},

number = C,

volume = 236,

place = {United States},

year = {2019},

month = {3}

}