tokens: that we assume to be fully ionised and composed of hydrogen and helium with the equilibration timescales and where and stand for the electron and ion mass, respectively; is the ion number density with the proton mass and being the ion mean molecular weight; is the ion charge number; the electron charge; and is the Coulomb logarithm. Since , we neglect the term in the energy exchange rates. For a mixture of hydrogen and helium, the timescale of the temperature coupling between hydrogen and helium would be proportional to . Thus, the equilibration between hydrogen and helium operates times faster than the equilibration timescale between hydrogen and electrons or helium and electrons. For hydrogen or helium, the ratio of also simplifies to (not the case for heavier elements). Therefore, protons and HeIII can be considered as one single population sharing the same temperature. Writing , we see that The variation in energy is symmetric between the transfer of ion energy and electron energy, but this is not the case for the variation in temperature, which is . Therefore, the difference in temperature change between ions and electrons will be a factor , where is the “mean molecular weight of electrons” . For astrophysical plasmas essentially composed of hydrogen and helium, the change in temperature between ions and electrons is close to symmetric. To solve for the system of two non-linear coupled equations, we rewrite ( 16 ) as where . We introduce the vector , and rewrite the previous two equations as The goal is now to iterate on to find the value of and as required by equations ( 20 ). The derivatives and are obtained by differentiating equations ( 20 ) with respect to and , where the superscript replaces the superscript in ( 20 ). The values of are obtained with Cramer’s rule, and is added to the value of until the variation in . 3. Numerical tests If not indicated, we use a courant factor of with adaptive time-stepping between the different levels of refinement. Fig. 1 Top : diffusion of a pressure step function as a function of position at (black), (red), (blue) for the simulation (symbols) and the analytic solution (solid lines). The extra levels of refinement are indicated as dashed lines. Bottom : relative errors of the simulation with respect to the analytic solution. 3.1. Diffusion of a step function We performed a simple 1D test of the diffusion of a step function with a constant diffusion coefficient of without hydrodynamics. The initial setup is a pressure of between and outside with . We start with a minimum level of refinement of and refine up to level wherever the relative pressure variation is larger than . The time step is chosen to be (for level 7), where is the stability time step for diffusion (only useful for explicit schemes). The time evolution of the step function has the following analytical solution for a constant diffusion coefficient where . Figure 1 shows the result of the diffusion of the step function with the
ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O