Linear stability analysis via simulated annealing and accelerated relaxation

TL;DR

This paper introduces a stability analysis method using simulated annealing with an accelerated relaxation technique, applied to low-beta reduced MHD, to efficiently find equilibria and assess stability in Hamiltonian systems.

Contribution

It develops an accelerated relaxation approach within simulated annealing for stability analysis of noncanonical Hamiltonian systems, including explicit formulation for low-beta reduced MHD.

Findings

Efficient stability analysis via SA with acceleration demonstrated.

Explicit formulation for low-beta reduced MHD provided.

Balance of advection fields is crucial for SA effectiveness.

Abstract

Simulated annealing (SA) is a kind of relaxation method for finding equilibria of Hamiltonian systems. A set of evolution equations is solved with SA, which is derived from the original Hamiltonian system so that the energy of the system changes monotonically while preserving Casimir invariants inherent to noncanonical Hamiltonian systems. The energy extremum reached by SA is an equilibrium. Since SA searches for an energy extremum, it can also be used for stability analysis when initiated from a state where a perturbation is added to an equilibrium. The procedure of the stability analysis is explained, and some examples are shown. Because the time evolution is computationally time consuming, efficient relaxation is necessary for SA to be practically useful. An acceleration method is developed by introducing time dependence in the symmetric kernel used in the double bracket, which is part of the SA formulation described here. An explicit formulation for low-beta reduced magnetohydrodynamics (MHD) in cylindrical geometry is presented. Since SA for low-beta reduced MHD has two advection fields that relax, it is important to balance the orders of magnitude of these advection fields.