Variational integration for ideal magnetohydrodynamics with built-in advection equations

TL;DR

This paper develops a variational integrator for ideal magnetohydrodynamics that preserves key physical properties and inherently incorporates advection equations, demonstrated through 2D simulations avoiding numerical reconnection.

Contribution

It introduces a novel discretization of Newcomb's Lagrangian using discrete exterior calculus, resulting in symplectic, momentum-preserving schemes with built-in advection equations.

Findings

Numerical reconnection is avoided in 2D simulations with singular current sheets.

The method successfully simulates the coalescence instability and finds relaxed equilibrium states.

The integrator preserves physical invariants and reduces numerical dissipation.

Abstract

Newcomb's Lagrangian for ideal magnetohydrodynamics (MHD) in Lagrangian labeling is discretized using discrete exterior calculus. Variational integrators for ideal MHD are derived thereafter. Besides being symplectic and momentum-preserving, the schemes inherit built-in advection equations from Newcomb's formulation, and therefore avoid solving them and the accompanying error and dissipation. We implement the method in 2D and show that numerical reconnection does not take place when singular current sheets are present. We then apply it to studying the dynamics of the ideal coalescence instability with multiple islands. The relaxed equilibrium state with embedded current sheets is obtained numerically.