# Huge accelerations in plasma simulation

I'm trying to make a numerical simulation of pulsar magnetosphere using FDTD on a log-spherical Yee lattice for fields and PIC for plasma particles. Field part is working like charm but issue arises when adding the plasma.

To simulate the dynamics of charged plasma particles I use Newton-Lorenz equation:

$\frac{d\gamma\vec{v}}{dt} = \frac{e}{m}\left( \vec{E} + \frac{\vec{v}}{c} \times \vec{B} \right)$

Adopting CGS system of units, characteristic values for $\vec{E}$ and $\vec{B}$ are of order of magnitule $10^{12}$, charge to mass ratio for electron is of order of magnitude $10^{17}$, so the value for acceleration is ~$10^{29}$.

In the code itself I normalized all variables in the following manner:

$\vec{u} = \frac{\gamma\vec{v}}{c}\\ \vec{E}^{*} = \vec{E}/B_0\\ \vec{B}^{*} = \vec{B}/B_0\\ \tau = \frac{ct}{R}\\$

Where $B_0=10^{12}$, $R=10^6$ (radius of the pulsar) and $c$ is the speed of light. As a result, my equation looks as follows:

$\frac{d\vec{u}}{dt} = \frac{eRB_0}{mc^2}\left(\vec{E}^{*}+\frac{\vec{u}\times\vec{B}^{*}}{\gamma}\right)$

Normalization factor $\frac{eRB_0}{mc^2}\sim 10^{15}$, acceleration is still huge.

I may choose different normalization for time as follows:

$t = t_0\tau\\ \frac{eB_0}{mc}t_0 = 1$

But with $\tau$ chosen in that way it takes literally forever for fields to get to stationary configuration.

What can I do?

• With fields that strong, you can't use the non-relativistic Newton-Lorenz equation. You must treat things relativistically in order to avoid diverging speeds. In addition, the field itself varies based on reference frame, and therefore varies with the speed of the particles. – probably_someone Feb 20 '18 at 3:46
• @probably_someone, I do use relativistic equation in the code, so Lorenz factor is there in my equations (and also lapse function due to curvature of space). I just omitted them here in the question so they don't distract people from the essence of the problem. Will it be better if I edit them in? – Tajimura Feb 20 '18 at 4:11
• If I recall correctly, there's a stack exchange just on scientific computing. This question might work better there? – Chris Feb 20 '18 at 4:26
• @Chris, I tried posting the question there, but was not able to do it, 'cause SE refuses to add tags in computational physics site. I dunno why it's so, never had that happen in Physics SE and Math SE. Any way to move the question there? – Tajimura Feb 20 '18 at 4:46
• @Tajimura I believe you can flag your post and ask a moderator to do it. – Chris Feb 20 '18 at 4:48