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A Parareal in time numerical method for the collisional Vlasov equation in the hyperbolic scaling

Tino Laidin, Thomas Rey

TL;DR

The paper addresses the high computational cost of solving scaled collisional Vlasov equations by introducing a multiscale parareal in time method that couples a cheap fluid coarse propagator with a high-fidelity kinetic fine propagator. It leverages Asymptotic-Preserving discretizations to remain stable for all Knudsen numbers $\varepsilon$, enabling accurate recovery of macroscopic moments. The authors demonstrate exponential convergence of the parareal iterations and report substantial speedups in 1D/3D test scenarios, validating the approach on Sod shock tubes, blast waves, and exterior-force cases. This framework enables efficient, parallel-in-time computation of kinetic moments and provides a foundation for MPI-scale extensions and higher-dimensional or Boltzmann-type extensions in future work.

Abstract

We present the design of a multiscale parareal method for kinetic equations in the fluid dynamic regime. The goal is to reduce the cost of a fully kinetic simulation using a parallel in time procedure. Using the multiscale property of kinetic models, the cheap, coarse propagator consists in a fluid solver and the fine (expensive) propagation is achieved through a kinetic solver for a collisional Vlasov equation. To validate our approach, we present simulations in the 1D in space, 3D in velocity settings over a wide range of initial data and kinetic regimes, showcasing the accuracy, efficiency, and the speedup capabilities of our method.

A Parareal in time numerical method for the collisional Vlasov equation in the hyperbolic scaling

TL;DR

The paper addresses the high computational cost of solving scaled collisional Vlasov equations by introducing a multiscale parareal in time method that couples a cheap fluid coarse propagator with a high-fidelity kinetic fine propagator. It leverages Asymptotic-Preserving discretizations to remain stable for all Knudsen numbers , enabling accurate recovery of macroscopic moments. The authors demonstrate exponential convergence of the parareal iterations and report substantial speedups in 1D/3D test scenarios, validating the approach on Sod shock tubes, blast waves, and exterior-force cases. This framework enables efficient, parallel-in-time computation of kinetic moments and provides a foundation for MPI-scale extensions and higher-dimensional or Boltzmann-type extensions in future work.

Abstract

We present the design of a multiscale parareal method for kinetic equations in the fluid dynamic regime. The goal is to reduce the cost of a fully kinetic simulation using a parallel in time procedure. Using the multiscale property of kinetic models, the cheap, coarse propagator consists in a fluid solver and the fine (expensive) propagation is achieved through a kinetic solver for a collisional Vlasov equation. To validate our approach, we present simulations in the 1D in space, 3D in velocity settings over a wide range of initial data and kinetic regimes, showcasing the accuracy, efficiency, and the speedup capabilities of our method.

Paper Structure

This paper contains 15 sections, 44 equations, 7 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Multiscale parareal algorithm
  3. Link between the scales
  4. Semi-discretization in time
  5. Numerical schemes
  6. Kinetic schemes
  7. Fluid schemes
  8. Discrete lifting and projection operators
  9. Parallelization of the method
  10. Numerical results
  11. Test 1: Sod shock tube
  12. Test 2: Blast waves
  13. Test 3: Exterior forces
  14. Conclusion
  15. MPI implementation of the multiscale parareal method

Figures (7)

  • Figure 1: Test 1 - Sod shock tube, $\varepsilon=10^{-2}$: Snapshots of the density (Top), $x$ mean velocity (Middle) and Temperature (Bottom) at times $T^n=0.05$ (Left), $0.15$ (Middle) and $0.25$ (Right).
  • Figure 2: Test 1 - Sod shock tube, $\varepsilon=10^{-2}$: Convergence of the successive errors.
  • Figure 3: Test 2 - Blast waves, $\varepsilon=10^{-2}$: Snapshots of the density (Top), $x$ mean velocity (Middle) and Temperature (Bottom) at times $T^n=0.1$ (Left), $0.23$ (Middle) and $0.3$ (Right).
  • Figure 4: Test 2 - Blast waves, $\varepsilon=10^{-2}$: Snapshots of the pointwise difference on the density (Top), $x$ mean velocity (Middle) and Temperature (Bottom) at times $T^n=0.1$ (Left), $0.23$ (Middle) and $0.3$ (Right).
  • Figure 5: Test 2 - Blast waves, $\varepsilon=10^{-2}$: Convergence of the successive errors.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Remark 2