Title: A collision-based hybrid method for the BGK equation

Journal Article · · Journal of Computational Physics
 [1]; ORCiD logo [2]; ORCiD logo [3]
  1. Yonsei University, Seoul (Korea, Republic of)
  2. Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
  3. University of Notre Dame, IN (United States)
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In this article, we apply the collision-based hybrid method introduced by Hauck and McClarren to the Boltzmann equation with the BGK operator and a hyperbolic scaling. An implicit treatment of the source term is used to handle stiffness associated with the BGK operator. Although it helps the numerical scheme become stable with a large time step size, it is still not obvious to achieve the desired order of accuracy due to the relationship between the size of the spatial cell and the mean free path. Without asymptotic preserving property, a very restricted grid size is required to resolve the mean free path, which is not practical. Our approaches are based on the noncollision-collision decomposition of the BGK equation. We introduce the arbitrary order of nodal discontinuous Galerkin (DG) discretization in space with a semi-implicit time-stepping method; we employ the backward Euler time integration for the uncollided equation and the 2nd order predictor-corrector scheme for the collided equation, i.e., both source terms in uncollided and collided equations are treated implicitly and only streaming term in the collided equation is solved explicitly. This improves the computational efficiency without the complexity of the numerical implementation. Numerical results are presented for various Knudsen numbers to present the effectiveness and accuracy of our hybrid method. Also, we compare the solutions of the hybrid and non-hybrid schemes.

Research Organization:
Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
Sponsoring Organization:
National Research Foundation of Korea (NRF); USDOE; USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
Grant/Contract Number:
AC05-00OR22725
OSTI ID:
2538487
Journal Information:
Journal of Computational Physics, Journal Name: Journal of Computational Physics Vol. 501; ISSN 0021-9991
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English

References (49)

Weak solutions of nonlinear hyperbolic equations and their numerical computation journal February 1954
Rarefied Flow Computations Using Nonlinear Model Boltzmann Equations journal September 1995
Discrete-Velocity Models and Numerical Schemes for the Boltzmann-BGK Equation in Plane and Axisymmetric Geometries journal August 2000
A Gas-Kinetic BGK Scheme for the Navier–Stokes Equations and Its Connection with Artificial Dissipation and Godunov Method journal July 2001
Conservative Numerical Schemes for the Vlasov Equation journal September 2001
Fluid dynamic limits of kinetic equations. I. Formal derivations journal April 1991
Generalization of the Krook kinetic relaxation equation journal January 1972
The BGK-model with velocity-dependent collision frequency journal February 1997
Implicit–Explicit Schemes for BGK Kinetic Equations journal January 2007
An Asymptotic Preserving Scheme for the ES-BGK Model of the Boltzmann Equation journal June 2010
On a Class of Implicit–Explicit Runge–Kutta Schemes for Stiff Kinetic Equations Preserving the Navier–Stokes Limit journal July 2017
Semi-implicit Hybrid Discrete $$\left( \text {H}^T_N\right) $$ Approximation of Thermal Radiative Transfer journal November 2021
Half-Space Problems for the Boltzmann Equation: A Survey journal June 2006
A Conservative, Entropic Multispecies BGK Model journal June 2017
A Consistent BGK Model with Velocity-Dependent Collision Frequency for Gas Mixtures journal September 2021
A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws journal April 1978
TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: One-dimensional systems journal September 1989
Efficient implementation of essentially non-oscillatory shock-capturing schemes, II journal July 1989
Global existence to the BGK model of Boltzmann equation journal November 1989
Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations journal November 1997
The Gaussian-BGK model of Boltzmann equation with small Prandtl number journal November 2000
Approximate Lax–Wendroff discontinuous Galerkin methods for hyperbolic conservation laws journal September 2017
Uniformly stable numerical schemes for the Boltzmann equation preserving the compressible Navier–Stokes asymptotics journal April 2008
Semi-implicit time integration for thermal radiative transfer journal August 2008
Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states journal April 2009
Microscopically implicit–macroscopically explicit schemes for the BGK equation journal January 2012
Energy-conserving discontinuous Galerkin methods for the Vlasov–Ampère system journal January 2014
High order asymptotic preserving nodal discontinuous Galerkin IMEX schemes for the BGK equation journal March 2015
An arbitrary-order, fully implicit, hybrid kinetic solver for linear radiative transport using integral deferred correction journal October 2017
Hybrid methods for radiation transport using diagonally implicit Runge–Kutta and space–time discontinuous Galerkin time integration journal January 2019
A fast implicit solver for semiconductor models in one space dimension journal September 2020
Improvements to a class of hybrid methods for radiation transport: Nyström reconstruction and defect correction methods journal December 2020
Conservative semi-Lagrangian schemes for kinetic equations Part II: Applications journal July 2021
Numerical schemes for a multi-species BGK model with velocity-dependent collision frequency journal January 2023
An electron conductivity model for dense plasmas journal January 1984
Multigroup Neutron Transport Using a Collision-Based Hybrid Method journal February 2023
Moment-Based Acceleration for Neutral Gas Kinetics with BGK Collision Operator journal July 2014
A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems journal May 1954
Highly Efficient Strong Stability-Preserving Runge–Kutta Methods with Low-Storage Implementations journal January 2008
Numerical Passage from Kinetic to Fluid Equations journal February 1991
A Collision-Based Hybrid Method for Time-Dependent, Linear, Kinetic Transport Equations journal January 2013
A Fast Solver for Implicit Integration of the Vlasov--Poisson System in the Eulerian Framework journal January 2018
Analysis of a New Implicit Solver for a Semiconductor Model journal January 2021
A Consistency Result for a Discrete-Velocity Model of the Boltzmann Equation journal October 1997
Discrete Velocity Model and Implicit Scheme for the bgk Equation of Rarefied gas Dynamics journal November 2000
A First-Collision Source Method That Satisfies Discrete S n Transport Balance journal June 1990
Boundary layers and homogenization of transport processes journal January 1979
A consistent kinetic model for a two-component mixture with an application to plasma journal January 2017
Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations journal March 2019

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97 MATHEMATICS AND COMPUTING
Asymptotic-preserving
BGK equation
Discontinuous Galerkin
Discrete velocity
Gas-injection
Hybrid decomposition