Energy behaviour of the Boris method for charged-particle dynamics
Ernst Hairer and Christian Lubich
Abstract. The Boris algorithm is a widely used numerical integrator for the motion of particles in
a magnetic field. This article proves near-conservation of energy over %exponentially
very long times in the special cases where the magnetic field is constant or
the electric potential is quadratic. When none of these
assumptions is satisfied, it is illustrated by numerical examples that the numerical energy
can have a linear drift or its error can behave like a random walk.
If the system has a rotational symmetry and the magnetic field is constant,
then also the momentum is approximately preserved over very long times,
but in a spatially varying magnetic field this is generally not satisfied.
Key Words. Boris algorithm,
charged particle, magnetic field,
energy conservation,
backward error analysis, modified differential equation.