Abstract
A modification of the standard Boris algorithm, called filtered Boris algorithm, is proposed for the numerical integration of the equations of motion of charged particles in a strong non-uniform magnetic field in the asymptotic scaling known as maximal ordering. With an appropriate choice of filters, second-order error bounds in the position and in the parallel velocity, and first-order error bounds in the normal velocity are obtained with respect to the scaling parameter. This also yields a second-order approximation to the guiding center motion. The proof compares the modulated Fourier expansions of the exact and the numerical solutions. Numerical experiments illustrate the error behaviour of the filtered Boris algorithm.
Highlights
In this paper we propose and analyse a numerical integrator for the equations of motion of a charged particle in a strong inhomogeneous magnetic field,B Christian LubichChina x(t) = x(t) × B(x(t), t) + E(x(t), t) with B(x, t) = 1 ε B0 (ε x ) +B1 (x, t) for 0 < ε (1.1)This scaling is of interest in particle methods in plasma physics and is called maximal ordering in [2]; see [12] for a careful discussion of scalings and a rigorous analysis of this model
Our main theoretical result in this paper is the following error bound for the filtered Boris algorithm
If in the filtered Boris algorithm, – xn is given by (2.4) with the function θ of (2.5), and – the filter functions and Υ are defined as in Algorithm 2.1, the errors in the positions and the velocities are bounded by xn − x(tn) = O(ε2), vn − v = O(ε2), v⊥n − v⊥(tn) = O(ε)
Summary
This scaling is of interest in particle methods in plasma physics and is called maximal ordering in [2]; see [12] for a careful discussion of scalings and a rigorous analysis of this model. We propose a modification, which we name filtered Boris algorithm This modified integrator allows us to obtain better accuracy with considerably larger time steps, at minor additional computational cost. 4 yield the motion of the guiding center up to O(ε2) They coincide up to O(ε2) with the guiding center equations of the numerical approximation given by the filtered Boris integrator for an appropriate filter and for non-resonant step sizes h ≤ Cε with a possibly large constant C. 7 we describe a related, but different integrator, called two-point filtered Boris algorithm, which evaluates the magnetic field both in the current position and in the current guiding center approximation in each step, and which has similar convergence properties to the previously considered filtered Boris method.
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