Uncertainty Quantification for the Homogeneous Landau-Fokker-Planck Equation via Deterministic Particle Galerkin methods

TL;DR

This paper introduces a deterministic particle method for solving the homogeneous Landau-Fokker-Planck equation with uncertainty, achieving spectral accuracy in uncertainty space while maintaining key physical properties.

Contribution

It presents a novel particle approximation based on gradient flow reformulation and spectral stochastic Galerkin methods, ensuring structural properties and regularity in the presence of uncertainty.

Findings

Spectral accuracy in uncertainty space.

Preservation of positivity, conservation, and entropy.

Validated through extensive numerical tests.

Abstract

We design a deterministic particle method for the solution of the spatially homogeneous Landau equation with uncertainty. The deterministic particle approximation is based on the reformulation of the Landau equation as a formal gradient flow on the set of probability measures, whereas the propagation of uncertain quantities is computed by means of a sg representation of each particle. This approach guarantees spectral accuracy in uncertainty space while preserving the fundamental structural properties of the model: the positivity of the solution, the conservation of invariant quantities, and the entropy production. We provide a regularity results for the particle method in the random space. We perform the numerical validation of the particle method in a wealth of test cases.