# American Institute of Mathematical Sciences

September  2017, 10(3): 643-668. doi: 10.3934/krm.2017026

## Asymptotic preserving and time diminishing schemes for rarefied gas dynamic

 1 INRIA-Rennes Bretagne Atlantique, IPSO Project, and IRMAR (Université de Rennes 1), 35042 Rennes, France 2 Department of Mathematics and Computer Science, University of Ferrara, 44121 Ferrara, Italy 3 CNRS and IRMAR (Université de Rennes 1 and ENS Rennes), and INRIA-Rennes Bretagne Atlantique, IPSO Project, 35042 Rennes, France

Received  June 2016 Revised  September 2016 Published  December 2016

Fund Project: The authors have been supported by the ANR project Moonrise.

In this work, we introduce a new class of numerical schemes for rarefied gas dynamic problems described by collisional kinetic equations. The idea consists in reformulating the problem using a micro-macro decomposition and successively in solving the microscopic part by using asymptotic preserving Monte Carlo methods. We consider two types of decompositions, the first leading to the Euler system of gas dynamics while the second to the Navier-Stokes equations for the macroscopic part. In addition, the particle method which solves the microscopic part is designed in such a way that the global scheme becomes computationally less expensive as the solution approaches the equilibrium state as opposite to standard methods for kinetic equations which computational cost increases with the number of interactions. At the same time, the statistical error due to the particle part of the solution decreases as the system approach the equilibrium state. This causes the method to degenerate to the sole solution of the macroscopic hydrodynamic equations (Euler or Navier-Stokes) in the limit of infinite number of collisions. In a last part, we will show the behaviors of this new approach in comparisons to standard Monte Carlo techniques for solving the kinetic equation by testing it on different problems which typically arise in rarefied gas dynamic simulations.

Citation: Nicolas Crouseilles, Giacomo Dimarco, Mohammed Lemou. Asymptotic preserving and time diminishing schemes for rarefied gas dynamic. Kinetic & Related Models, 2017, 10 (3) : 643-668. doi: 10.3934/krm.2017026
##### References:

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##### References:
Density, velocity and temperature profiles (from top to bottom) for: left Asymptotic Preserving Time Diminishing Navier-Stokes method, right Monte Carlo method, $\varepsilon=10^{-2}$. The dotted line is a reference solution computed with a deterministic DVM method while the continuous line is a reference solution for the compressible Euler equations. Unsteady Shock test
Density, velocity and temperature profiles (from top to bottom) for: left Asymptotic Preserving Time Diminishing Navier-Stokes method, right Monte Carlo method, $\varepsilon=10^{-3}$. The dotted line is a reference solution computed with a deterministic DVM method while the continuous line is a reference solution for the compressible Euler equations. Unsteady Shock test
Density, velocity and temperature profiles (from top to bottom) for: left Asymptotic Preserving Time Diminishing Navier-Stokes method, right Monte Carlo method, $\varepsilon=10^{-4}$. The dotted line is a reference solution computed with a deterministic DVM method while the continuous line is a reference solution for the compressible Euler equations. Unsteady Shock test
Left: time evolution of the number of effective particles (in semi-logarithmic scale) used in the Asymptotic Preserving Time Diminishing (Euler) method and in the Monte Carlo method for different values of the Knudsen number ($\varepsilon=10^{-2}, 5\cdot 10^{-3}, 10^{-3}, 5\cdot 10^{-4}, 10^{-4}$). Middle: time evolution of the ratio of the number of particles used for APTD versus the number of particles for the corresponding MC simulation for different values of $\varepsilon$. Right: time evolution of the ratio of the number of particles used for APTD versus the number of particles for APTDNS for different values of $\varepsilon$. Unsteady Shock test
Density, velocity and temperature profiles (from top to bottom) for: left Asymptotic Preserving Time Diminishing Navier-Stokes method, right Monte Carlo method, $\varepsilon=10^{-2}$. The dotted line is a reference solution computed with a deterministic DVM method while the continuous line is a reference solution for the compressible Euler equations. Sod test
Density, velocity and temperature profiles (from top to bottom) for: left Asymptotic Preserving Time Diminishing Navier-Stokes method, right Monte Carlo method, $\varepsilon=10^{-3}$. The dotted line is a reference solution computed with a deterministic DVM method while the continuous line is a reference solution for the compressible Euler equations. Sod test
Density, velocity and temperature profiles (from top to bottom) for: left Asymptotic Preserving Time Diminishing Navier-Stokes method, right Monte Carlo method, $\varepsilon=10^{-4}$. The dotted line is a reference solution computed with a deterministic DVM method while the continuous line is a reference solution for the compressible Euler equations. Sod test
Left: time evolution of the number of effective particles (in semi-logarithmic scale) used in the Asymptotic Preserving Time Diminishing (Euler) method and in the Monte Carlo method for different values of the Knudsen number ($\varepsilon=10^{-2}, 5\cdot 10^{-3}, 10^{-3}, 5\cdot 10^{-4}, 10^{-4}$). Middle: time evolution of the ratio of the number of particles used for APTD versus the number of particles for the corresponding MC simulation for different values of $\varepsilon$. Right: time evolution of the ratio of the number of particles used for APTD versus the number of particles for APTDNS for different values of $\varepsilon$. Sod test
Error ($L^1$ norm) for the density, the mean velocity and the temperature for the Asymptotic Preserving Time Diminishing NS method (left column), and for the Monte Carlo method (right column) for different values of $\varepsilon$ (from top to bottom, $\varepsilon =10^{-2}, 10^{-3}, 10^{-4}$)
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