A more accurate numerical scheme for diffusive shock acceleration

TL;DR

This paper introduces a more precise numerical method for modeling diffusive shock acceleration of cosmic rays, especially effective in scenarios with steep diffusivity gradients, improving accuracy over previous schemes.

Contribution

A new numerical scheme based on Stochastic Differential Equations that enhances accuracy in modeling cosmic ray acceleration at shocks with variable diffusivity.

Findings

The scheme outperforms earlier Cauchy-Euler schemes in steep gradient scenarios.

It provides comparable results to existing methods for constant diffusivity.

Demonstrated improved accuracy using analytical flow profiles with finite shock width.

Abstract

We present a more accurate numerical scheme for the calculation of diffusive shock acceleration of cosmic rays using Stochastic Differential Equations. The accuracy of this scheme is demonstrated using a simple analytical flow profile that contains a shock of finite width and a varying diffusivity of the cosmic rays, where the diffusivity decreases across the shock. We compare the results for the slope of the momentum distribution with those obtained from a perturbation analysis valid for finite but small shock width. These calculations show that this scheme, although computationally more expensive, provides a significantly better performance than the Cauchy-Euler type schemes that were proposed earlier in the case where steep gradients in the cosmic ray diffusivity occur. For constant diffusivity the proposed scheme gives similar results as the Cauchy-Euler scheme.