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Title: 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 » investigate the accuracy and efficiency of the numerical schemes for several numerical test cases, including a steady-state shock-structure problem.« less

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}
}
  • We compute, for the first time, high-order entropy-based (more » $$M_N$$) models for a linear transport equation on a one-dimensional, slab geometry. We simulate two test problems from the literature: the two-beam instability and the plane-source problem. In the former case we compute solutions for systems up to order $N=5$ ; in the latter, up to $N=15$. The most notable outcome of these results is the existence of shocks in the steady-state profiles of the two-beam instability for all odd values of $N$.« less
  • We present a numerical algorithm to implement entropy-based (M{sub N}) moment models in the context of a simple, linear kinetic equation for particles moving through a material slab. The closure for these models - as is the case for all entropy-based models - is derived through the solution of constrained, convex optimization problem. The algorithm has two components. The first component is a discretization of the moment equations which preserves the set of realizable moments, thereby ensuring that the optimization problem has a solution (in exact arithmetic). The discretization is a second-order kinetic scheme which uses MUSCL-type limiting in spacemore » and a strong-stability-preserving, Runge-Kutta time integrator. The second component of the algorithm is a Newton-based solver for the dual optimization problem, which uses an adaptive quadrature to evaluate integrals in the dual objective and its derivatives. The accuracy of the numerical solution to the dual problem plays a key role in the time step restriction for the kinetic scheme. We study in detail the difficulties in the dual problem that arise near the boundary of realizable moments, where quadrature formulas are less reliable and the Hessian of the dual objection function is highly ill-conditioned. Extensive numerical experiments are performed to illustrate these difficulties. In cases where the dual problem becomes 'too difficult' to solve numerically, we propose a regularization technique to artificially move moments away from the realizable boundary in a way that still preserves local particle concentrations. We present results of numerical simulations for two challenging test problems in order to quantify the characteristics of the optimization solver and to investigate when and how frequently the regularization is needed.« less
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