Magnetospheric Plasma Physics: The Impact of Jim Dungey’s Research by David Southwood Stanley W. H. Cowley FRS & Simon Mitton

Author:David Southwood, Stanley W. H. Cowley FRS & Simon Mitton
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

6.1 Introduction

This paper addresses complexity in many-body calculations. Investigation by computer simulation was in its infancy when I was one of Jim Dungey’s students, but is now used in almost all areas of science and engineering. Jim initially pointed me to computational studies looking at non-adiabatic ion trajectories and their effects on currents in the current sheet of the geomagnetic tail (Eastwood 1972, 1974); this is where I first became involved in many-body calculations, using particle methods to model the interactions of plasma with electric and magnetic fields.

Simulation studies face a compromise between the detail of the model and computational resources. In principle classical and pseudo-classical systems may be described in terms of positions, velocities and force laws of the particles of which it is composed. Unfortunately, the vast number of particles involved in quite simple situations generally make such a detailed description computationally impractical, despite the enormous growth in computer power over the past four decades. I learnt from Jim the importance of assessing length- and time-scales and using them to reduce the mathematical models (e.g., as in hydromagnetics and kinetic plasma models). Applying this principle to many-body calculations can lead to models that are sufficiently detailed to reproduce important physical effects but not too detailed to make calculations impracticable.

Even with reduced models, a large part of the computational work in many-body calculations is the evaluation of interparticle forces. For example, given N charged particles, the Coulombic field at one particle has N-1 contributions from the other particles, so to compute fields at all N particles has complexity O(N2). This paper reviews methods of reducing this complexity to O(N log N) in different situations. Reduced to its basics, the problem is to evaluate in O(N log N) operations the convolution sum or integral:

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