Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

High-performance computing for the BGK model of the Boltzmann equation with a meshfree Arbitrary Lagrangian-Eulerian (ALE) method

Panchatchram Mariappan, Klaas Willems, Gangadhara Boregowda, Sudarshan Tiwari, Axel Klar

TL;DR

This paper presents high-performance computing for the BGK model of the Boltzmann equation with a mesh-free method using an Arbitrary-Lagrangian-Eulerian (ALE) method, where the approximation of spatial derivatives and the reconstruction of a function is based on the weighted least squares method.

Abstract

In this paper, we present high-performance computing for the BGK model of the Boltzmann equation with a mesh-free method. For the numerical simulation of the BGK equation we use an Arbitrary-Lagrangian-Eulerian (ALE) method developed in previous work, where the approximation of spatial derivatives and the reconstruction of a function is based on the weighted least squares method. A Graphics Processing Unit (GPU) is used to accelerate the code. The result is compared with sequential and parallel versions of the CPU code. Two and three-dimensional driven cavity problems are solved, where a speed-up of several orders of magnitude is obtained compared to a sequential CPU simulation.

High-performance computing for the BGK model of the Boltzmann equation with a meshfree Arbitrary Lagrangian-Eulerian (ALE) method

TL;DR

This paper presents high-performance computing for the BGK model of the Boltzmann equation with a mesh-free method using an Arbitrary-Lagrangian-Eulerian (ALE) method, where the approximation of spatial derivatives and the reconstruction of a function is based on the weighted least squares method.

Abstract

In this paper, we present high-performance computing for the BGK model of the Boltzmann equation with a mesh-free method. For the numerical simulation of the BGK equation we use an Arbitrary-Lagrangian-Eulerian (ALE) method developed in previous work, where the approximation of spatial derivatives and the reconstruction of a function is based on the weighted least squares method. A Graphics Processing Unit (GPU) is used to accelerate the code. The result is compared with sequential and parallel versions of the CPU code. Two and three-dimensional driven cavity problems are solved, where a speed-up of several orders of magnitude is obtained compared to a sequential CPU simulation.
Paper Structure (24 sections, 49 equations, 5 figures, 3 tables)

This paper contains 24 sections, 49 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The BGK model for rarefied gas dynamics
  3. Chu-reduction
  4. Numerical schemes
  5. ALE formulation
  6. ALE formulation for original model
  7. ALE for reduced model
  8. Time discretization
  9. First order time splitting scheme for original model
  10. Time splitting scheme for the reduced model
  11. Velocity discretization
  12. Spatial discretization
  13. Approximation of spatial derivatives
  14. First order upwind scheme
  15. Particle management
  16. ...and 9 more sections

Figures (5)

  • Figure 5.1: Two dimensional driven cavity velocity vectors for Knudsen numbers and Reynolds numbers $Kn = 1.1, Re = 2.968 \times 10^{-4}$ (left) and $Kn = 0.11, Re = 2.968 \times 10^{-5}$ (right).
  • Figure 5.2: Three dimensional driven cavity velocity vectors for Knudsen numbers and Reynolds numbers $Kn = 1.1, Re = 2.968 \times 10^{-4}$ (left) and $Kn = 0.11, Re = 2.968 \times 10^{-5}$ (right).
  • Figure 5.3: Three dimensional velocity vectors in $xz$ plane at half of the $y$-axis for Knudsen numbers and Reynolds numbers $Kn = 1.1, Re = 2.968 \times 10^{-4}$ (left) and $Kn = 0.11, Re = 2.968 \times 10^{-5}$ (right).
  • Figure 5.4: Comparison of GPU, OMP and CPU time for $3D$ driven cavity problem for different numbers of initial spatial grid points for fixed velocity grids $N_v = 15$ at final time $t_{final} = 400\times\Delta t$ .
  • Figure 5.5: Comparison of GPU, OMP and CPU time for $3D$ driven cavity problem for different numbers of velocity grid points for fixed number of initial spatial grid points $N = 40^3$ at final time $t_{final} = 400\times\Delta t$ .