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A convergent interacting particle method for computing KPP front speeds in random flows

Tan Zhang, Zhongjian Wang, Jack Xin, Zhiwen Zhang

TL;DR

This work develops a mesh-free, convergent interacting particle method (IPM) to compute Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in random, divergence-free flows by recasting the problem as evaluating the principal Lyapunov exponent $\mu(\lambda\bm e)$ of a linear advection-diffusion operator. It combines a random Fourier representation of the velocity field with a normalized Feynman-Kac semigroup and an Euler–Maruyama based time discretization, coupled via a multinomial-resampling particle system to estimate $\mu$ and hence $c^*(\bm z)$. The paper provides rigorous convergence and error results for the random Fourier approximation, operator splitting, and the IPM itself, and validates the approach against semi-Lagrangian methods in 2D and 3D flows, including 3D ABC and cellular configurations under random perturbations. The method demonstrates clear advantages in high dimensions and unbounded domains, including a dynamic shift technique to realize invariant-measure convergence on $\mathbb{R}^d$ and favorable computational scaling compared to Eulerian schemes.

Abstract

We aim to efficiently compute spreading speeds of reaction-diffusion-advection (RDA) fronts in divergence free random flows under the Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We study a stochastic interacting particle method (IPM) for the reduced principal eigenvalue (Lyapunov exponent) problem of an associated linear advection-diffusion operator with spatially random coefficients. The Fourier representation of the random advection field and the Feynman-Kac (FK) formula of the principal eigenvalue (Lyapunov exponent) form the foundation of our method implemented as a genetic evolution algorithm. The particles undergo advection-diffusion, and mutation/selection through a fitness function originated in the FK semigroup. We analyze convergence of the algorithm based on operator splitting, present numerical results on representative flows such as 2D cellular flow and 3D Arnold-Beltrami-Childress (ABC) flow under random perturbations. The 2D examples serve as a consistency check with semi-Lagrangian computation. The 3D results demonstrate that IPM, being mesh free and self-adaptive, is simple to implement and efficient for computing front spreading speeds in the advection-dominated regime for high-dimensional random flows on unbounded domains where no truncation is needed.

A convergent interacting particle method for computing KPP front speeds in random flows

TL;DR

This work develops a mesh-free, convergent interacting particle method (IPM) to compute Kolmogorov-Petrovsky-Piskunov (KPP) front speeds in random, divergence-free flows by recasting the problem as evaluating the principal Lyapunov exponent of a linear advection-diffusion operator. It combines a random Fourier representation of the velocity field with a normalized Feynman-Kac semigroup and an Euler–Maruyama based time discretization, coupled via a multinomial-resampling particle system to estimate and hence . The paper provides rigorous convergence and error results for the random Fourier approximation, operator splitting, and the IPM itself, and validates the approach against semi-Lagrangian methods in 2D and 3D flows, including 3D ABC and cellular configurations under random perturbations. The method demonstrates clear advantages in high dimensions and unbounded domains, including a dynamic shift technique to realize invariant-measure convergence on and favorable computational scaling compared to Eulerian schemes.

Abstract

We aim to efficiently compute spreading speeds of reaction-diffusion-advection (RDA) fronts in divergence free random flows under the Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We study a stochastic interacting particle method (IPM) for the reduced principal eigenvalue (Lyapunov exponent) problem of an associated linear advection-diffusion operator with spatially random coefficients. The Fourier representation of the random advection field and the Feynman-Kac (FK) formula of the principal eigenvalue (Lyapunov exponent) form the foundation of our method implemented as a genetic evolution algorithm. The particles undergo advection-diffusion, and mutation/selection through a fitness function originated in the FK semigroup. We analyze convergence of the algorithm based on operator splitting, present numerical results on representative flows such as 2D cellular flow and 3D Arnold-Beltrami-Childress (ABC) flow under random perturbations. The 2D examples serve as a consistency check with semi-Lagrangian computation. The 3D results demonstrate that IPM, being mesh free and self-adaptive, is simple to implement and efficient for computing front spreading speeds in the advection-dominated regime for high-dimensional random flows on unbounded domains where no truncation is needed.
Paper Structure (29 sections, 17 theorems, 126 equations, 18 figures, 1 algorithm)

This paper contains 29 sections, 17 theorems, 126 equations, 18 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Efficient IPM for KPP front speeds
  3. Random fields realization
  4. Computing $\mu$ by FK semigroup
  5. Numerical discretization and resampling techniques
  6. Convergence analysis
  7. Approximating Gaussian random field with a prescribed discrete spectrum by random Fourier method
  8. Approximating parabolic evolution by operator splitting
  9. Error estimate of particle method
  10. Error estimate in computing $\mu$ via random Fourier method
  11. Numerical experiments
  12. Convergence test in computing $\mu$
  13. IPM vs. Semi-Lagrangian computation of KPP front speed
  14. 2D cellular flow
  15. 2D random shear flow
  16. ...and 14 more sections

Key Result

Proposition 2.1

For any $m \in \mathcal{P}(\mathcal{D})$ and $\phi \in S$, there exists $C_{\mathcal{A}} > 0$ such that where $\delta_{c}=\inf\{\mu(\lambda)-\Re(z):z\in \sigma(\mathcal{A}) \setminus\{\mu(\lambda)\}\}>0$ is the spectral gap of the operator $\mathcal{A}$, $m_{c}$ is the invariant measure of $\Phi^{c}_{t}$.

Figures (18)

  • Figure 1: Performance of the coefficient of the generated random velocity field.
  • Figure 2: Front speed $c^*$ vs. $t$ of random shear flow with different Fourier modes $N_F$.
  • Figure 3: Computation of $c^*$ in 2D cellular flow.
  • Figure 4: Comparison of $c^*$ by IPM and SL+CN method.
  • Figure 5: Loglog plot and table of mesh size $N_x$ vs. Running time of SL+CN method in 3D Cellular flow (* in the table means the time is estimated).
  • ...and 13 more figures

Theorems & Definitions (26)

  • Proposition 2.1
  • Proposition 2.2: Lemma 3.6 in Junlong2022
  • Lemma 3.1
  • Proof 1
  • Lemma 3.2
  • Proposition 3.1
  • Proposition 3.2: Theorem 3.7 in Junlong2022
  • Proposition 3.3: Theorem 3.8 in Junlong2022
  • Proposition 3.4: Theorem 3.11 in Junlong2022
  • Proposition 3.5
  • ...and 16 more