# Efficient algorithms and implementations of entropy-based moment closures for rarefied gases

## Abstract

We present efficient algorithms and implementations of the 35-moment system equipped with the maximum-entropy closure in the context of rarefied gases. While closures based on the principle of entropy maximization have been shown to yield very promising results for moderately rarefied gas flows, the computational cost of these closures is in general much higher than for closure theories with explicit closed-form expressions of the closing fluxes, such as Grad's classical closure. Following a similar approach as Garrett et al. (2015) , we investigate efficient implementations of the computationally expensive numerical quadrature method used for the moment evaluations of the maximum-entropy distribution by exploiting its inherent fine-grained parallelism with the parallelism offered by multi-core processors and graphics cards. We show that using a single graphics card as an accelerator allows speed-ups of two orders of magnitude when compared to a serial CPU implementation. To accelerate the time-to-solution for steady-state problems, we propose a new semi-implicit time discretization scheme. The resulting nonlinear system of equations is solved with a Newton type method in the Lagrange multipliers of the dual optimization problem in order to reduce the computational cost. Additionally, fully explicit time-stepping schemes of first and second order accuracy are presented. Wemore »

- Authors:

- Publication Date:

- OSTI Identifier:
- 22622300

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 340; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY; ACCELERATORS; ACCURACY; ALGORITHMS; CLOSURES; COMPARATIVE EVALUATIONS; DISTRIBUTION; ENTROPY; EQUATIONS; EQUILIBRIUM; GAS FLOW; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; QUADRATURES; RAREFIED GASES; STEADY-STATE CONDITIONS; VELOCITY

### Citation Formats

```
Schaerer, Roman Pascal, E-mail: schaerer@mathcces.rwth-aachen.de, Bansal, Pratyuksh, and Torrilhon, Manuel.
```*Efficient algorithms and implementations of entropy-based moment closures for rarefied gases*. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.02.064.

```
Schaerer, Roman Pascal, E-mail: schaerer@mathcces.rwth-aachen.de, Bansal, Pratyuksh, & Torrilhon, Manuel.
```*Efficient algorithms and implementations of entropy-based moment closures for rarefied gases*. United States. doi:10.1016/J.JCP.2017.02.064.

```
Schaerer, Roman Pascal, E-mail: schaerer@mathcces.rwth-aachen.de, Bansal, Pratyuksh, and Torrilhon, Manuel. Sat .
"Efficient algorithms and implementations of entropy-based moment closures for rarefied gases". United States.
doi:10.1016/J.JCP.2017.02.064.
```

```
@article{osti_22622300,
```

title = {Efficient algorithms and implementations of entropy-based moment closures for rarefied gases},

author = {Schaerer, Roman Pascal, E-mail: schaerer@mathcces.rwth-aachen.de and Bansal, Pratyuksh and Torrilhon, Manuel},

abstractNote = {We present efficient algorithms and implementations of the 35-moment system equipped with the maximum-entropy closure in the context of rarefied gases. While closures based on the principle of entropy maximization have been shown to yield very promising results for moderately rarefied gas flows, the computational cost of these closures is in general much higher than for closure theories with explicit closed-form expressions of the closing fluxes, such as Grad's classical closure. Following a similar approach as Garrett et al. (2015) , we investigate efficient implementations of the computationally expensive numerical quadrature method used for the moment evaluations of the maximum-entropy distribution by exploiting its inherent fine-grained parallelism with the parallelism offered by multi-core processors and graphics cards. We show that using a single graphics card as an accelerator allows speed-ups of two orders of magnitude when compared to a serial CPU implementation. To accelerate the time-to-solution for steady-state problems, we propose a new semi-implicit time discretization scheme. The resulting nonlinear system of equations is solved with a Newton type method in the Lagrange multipliers of the dual optimization problem in order to reduce the computational cost. Additionally, fully explicit time-stepping schemes of first and second order accuracy are presented. We investigate the accuracy and efficiency of the numerical schemes for several numerical test cases, including a steady-state shock-structure problem.},

doi = {10.1016/J.JCP.2017.02.064},

journal = {Journal of Computational Physics},

number = ,

volume = 340,

place = {United States},

year = {Sat Jul 01 00:00:00 EDT 2017},

month = {Sat Jul 01 00:00:00 EDT 2017}

}