Source file twofluid.M

cat twofluid.M | m4

• Subroutine steptwofluid:

The subroutine steptwofluid advances all quantities from t to t+dt according to the two-fluid equations. As time stepping algorithm a trapezoidal leapfrog scheme is used. One iteration consist of the following steps (only the first half of a leapfrog time step is shown):
1. the quantity on the right hand side of the evolution equation for B' is calculated (compare the twofluid equations):

E' = (j x B' - j_i x B - grad p_e)/n = [(curl B) x (1-d_e² div grad) B - j_i x B - grad p_e)]/n
2. B' is calculated from B:  B' =  (1-d_e² div grad) B
3. B' is advanced from t to t+dt using dB'/dt = E'
4. B at t+dt is calculated by inverting the Helmholtz equation  (1-d_e² div grad) B = B', call to subroutine helmholtz (source file multigrid.M)
5. the ion current density is advanced from t to t+dt according to  dj_i/dt = -div (j_i j_i /n) + j x B - grad (p_e+p_i)
6. step density, electron and ion pressure
7. apply fourth order viscosities to B and j, density, electron and ion pressure
• Subroutine divergence:

The subroutine divergence is called to fix the divergence of the magnetic field (i.e. div B = 0) at start up and uses the same algorithm as described in the full-particle version of the subroutine divergence in the source file maxwell.M.

• isothermal: If the switch #define isothermal is used electron and ion pressure are determined from the density and the constant temperatures T_e and T_i.
• n_base: Since in the two-fluid equations the density appears at several location in theisothermal: If the switche #define isothermal is used the electron pressure is determined from the density and the constant electron temperature T_e. denominator numerical problems arise if the density becomes too small. To relax these problems an artificial homogeneous background density can be added. The parameter n_base sets the small artificial background density, for example #define n_base 0.1