Mimetic finite difference schemes for transport operators with divergence-free advective field and applications to plasma physics

TL;DR

This paper introduces a mimetic finite difference scheme on staggered grids for divergence-free transport operators, ensuring energy preservation and stability, with applications to plasma wave modeling in tokamak-like magnetic fields.

Contribution

The paper develops a new mimetic finite difference method that mimics divergence theorem preservation and energy conservation for wave problems with divergence-free velocity fields.

Findings

The scheme preserves energy in wave propagation simulations.

Application to plasma shear Alfvén waves demonstrates stability.

Method is versatile for divergence-free flow problems.

Abstract

In wave propagation problems, finite difference methods implemented on staggered grids are commonly used to avoid checkerboard patterns and to improve accuracy in the approximation of short-wavelength components of the solutions. In this study, we develop a mimetic finite difference (MFD) method on staggered grids for transport operators with divergence-free advective field that is proven to be energy-preserving in wave problems. This method mimics some characteristics of the summation-by-parts (SBP) operators framework, in particular it preserves the divergence theorem at the discrete level. Its design is intended to be versatile and applicable to wave problems characterized by a divergence-free velocity. As an application, we consider the electrostatic shear Alfv\'en waves (SAWs), appearing in the modeling of plasmas. These waves are solved in a magnetic field configuration recalling that of a tokamak device. The study of the generalized eigenvalue problem associated with the SAWs shows the energy conservation of the discretization scheme, demonstrating the stability of the numerical solution.