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A space-time hybrid parareal method for kinetic equations in the diffusive scaling

Tino Laidin

TL;DR

This work addresses the computational bottleneck of kinetic simulations in the diffusive scaling by formulating a space–time multiscale method that couples asymptotic-preserving micro–macro solvers with dynamic domain adaptation and a parareal in time algorithm. A macroscopic hierarchy from Chapman–Enskog expansion and two domain indicators guide adaptive switching between kinetic and fluid descriptions, while lifting connects macroscopic densities to kinetic perturbations. The coarse solver uses a fluid model and the fine solver uses a hybrid kinetic–fluid scheme, enabling parallel-in-time corrections and significant speedups across kinetic and fluid regimes, with convergence to machine precision and strong mass conservation properties. The approach yields substantial practical gains, is AP across regimes, and opens avenues for extension to higher dimensions, more complex collisions, and distributed-memory computation.

Abstract

We present a novel multiscale numerical approach that combines parallel-in-time computation with hybrid domain adaptation for linear collisional kinetic equations in the diffusive regime. The method addresses the computational challenges of kinetic simulations by integrating two complementary strategies: a parareal temporal parallelization method and a dynamic spatial domain adaptation based on perturbative analysis. The parallel in time approach employs a coarse fluid solver for efficient temporal propagation coupled with a fine, spatially-hybridized, kinetic solver for accurate resolution. Domain adaptation is governed by two criteria: one measuring the deviation from local velocity equilibrium, and another based on macroscopic quantities available throughout the computational domain. An asymptotic preserving micro-macro decomposition framework handles the stiffness of the original problem. This fully hybrid methodology significantly reduces computational costs compared to full kinetic approaches by exploiting the lower dimensionality of asymptotic fluid models while maintaining accuracy through selective kinetic resolution. The method demonstrates substantial speedup capabilities and efficiency gains across various kinetic regimes.

A space-time hybrid parareal method for kinetic equations in the diffusive scaling

TL;DR

This work addresses the computational bottleneck of kinetic simulations in the diffusive scaling by formulating a space–time multiscale method that couples asymptotic-preserving micro–macro solvers with dynamic domain adaptation and a parareal in time algorithm. A macroscopic hierarchy from Chapman–Enskog expansion and two domain indicators guide adaptive switching between kinetic and fluid descriptions, while lifting connects macroscopic densities to kinetic perturbations. The coarse solver uses a fluid model and the fine solver uses a hybrid kinetic–fluid scheme, enabling parallel-in-time corrections and significant speedups across kinetic and fluid regimes, with convergence to machine precision and strong mass conservation properties. The approach yields substantial practical gains, is AP across regimes, and opens avenues for extension to higher dimensions, more complex collisions, and distributed-memory computation.

Abstract

We present a novel multiscale numerical approach that combines parallel-in-time computation with hybrid domain adaptation for linear collisional kinetic equations in the diffusive regime. The method addresses the computational challenges of kinetic simulations by integrating two complementary strategies: a parareal temporal parallelization method and a dynamic spatial domain adaptation based on perturbative analysis. The parallel in time approach employs a coarse fluid solver for efficient temporal propagation coupled with a fine, spatially-hybridized, kinetic solver for accurate resolution. Domain adaptation is governed by two criteria: one measuring the deviation from local velocity equilibrium, and another based on macroscopic quantities available throughout the computational domain. An asymptotic preserving micro-macro decomposition framework handles the stiffness of the original problem. This fully hybrid methodology significantly reduces computational costs compared to full kinetic approaches by exploiting the lower dimensionality of asymptotic fluid models while maintaining accuracy through selective kinetic resolution. The method demonstrates substantial speedup capabilities and efficiency gains across various kinetic regimes.

Paper Structure

This paper contains 15 sections, 1 theorem, 29 equations, 8 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Dynamic domain adaptation
  3. Macroscopic hierarchy
  4. Domain indicators
  5. Numerical schemes
  6. Multiscale parareal algorithm
  7. Parallel in time integration
  8. Linking the scales
  9. Multiscale algorithm
  10. Implementation
  11. Numerical results
  12. Convergence
  13. Performances
  14. Mass conservation
  15. Conclusion

Key Result

Lemma 1

Let $(\rho_i^{n,k})_{(n,i)\in\mathcal{I}\times\{0,\dots,N_g\}}$ be given by Algorithm algo:paraMS and assume exact mass conservation of the hybrid numerical method eq:SHMM. We denote the mass of the system at time $t^n$ and parareal iteration $k$ as Then, for all $k>0$ and all $n\in\{0,\dots,N_g\}$,

Figures (8)

  • Figure 1: Asymptotic-Preserving diagram
  • Figure 2: Density snapshots for $\varepsilon=0.5$ obtained with Algorithm \ref{['algo:paraMS']}.
  • Figure 3: Electric field snapshots for $\varepsilon=0.5$ obtained with Algorithm \ref{['algo:paraMS']}.
  • Figure 4: Density snapshots for $\varepsilon=10^{-4}$ obtained with Algorithm \ref{['algo:paraMS']}.
  • Figure 5: Electric field snapshots for $\varepsilon=10^{-4}$ obtained with Algorithm \ref{['algo:paraMS']}.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Definition 1
  • Lemma 1: Conservation of mass
  • proof
  • Remark 1