When employing numeric schemes regarding derivatives of some values, relative accuracy is an important factor to be considered for the results to have any meaning. The numeric integration scheme (roughly) works as follows:

Assume a radial one-dimensional grid with grid points at heliocentric positions \(r_{i}\), with \(i\in[0,...,N-1]\), \(N\) number of grid points. All physical quantities are computed at the grid points corresponding to heliocentric positions above the photosphere. Let us assume one such quantity to be \(\psi_{i}\), where i again denotes the quantity stored (or computed) at grid point i.

Starting from an initial value \(\psi_{i}\) somewhere in the middle of the grid (e.g. the Alfvén point), one can compute the derivative \(\psi_{i}^{\prime}=\frac{\partial\psi}{\partial r}\) employing other physical quantities at that same grid point \(i\) (e.g. magnetic flux density \(B_{0,i}\), plasma mass density \(\rho_{i}\) and mean plasma flow velocity \(u_{i}\)).

Employing Euler’s method of integration (or Runke-Kutta of first order, RK1), a value for

at the next grid point \(i\pm 1\) may be obtained. The RK4-method works in a similar manner, and that one has been utilized in ZEPHYR. So in order of brevity, we will stick to the simpler RK1-method to demonstrate the problem.

In order to capture highly dynamic processes, the ”change in \(\psi\)” should be very small. We define the relative accuracy \(\epsilon\) of the integration process described by \ref{equ:euler} to be

The computations were parametrized to stay below a relative accuracy of \(\epsilon=10^{-9}\), which seems to be a sound value but is computationally unfeasable in the present setting.

Question: Is this the relative accuracy described in section 5.3 of \cite{Cranmer_2005}?