Metriplectic Integrators for the Landau Collision Operator

TL;DR

This paper introduces a new finite element-based framework leveraging the metriplectic structure to discretize the Landau collision operator, ensuring conservation laws and entropy production in the fully discrete system.

Contribution

It develops a general Galerkin discretization approach that preserves physical invariants and entropy behavior for the Landau collision integral.

Findings

Conservation of energy, momentum, and particle densities is algebraically demonstrated.

Entropy production is guaranteed in the fully discrete system.

The approach is mesh-independent and applicable to various finite element discretizations.

Abstract

We present a novel framework for addressing the nonlinear Landau collision integral in terms of finite element and other subspace projection methods. We employ the underlying metriplectic structure of the Landau collision integral and, using a Galerkin discretization for the velocity space, we transform the infinite-dimensional system into a finite-dimensional, time-continuous metriplectic system. Temporal discretization is accomplished using the concept of discrete gradients. The conservation of energy, momentum, and particle densities, as well as the production of entropy is demonstrated algebraically for the fully discrete system. Due to the generality of our approach, the conservation properties and the monotonic behavior of entropy are guaranteed for finite element discretizations in general, independently of the mesh configuration.