Source file hybrid.M

cat hybrid.M | m4

• Subroutine stephybrid:

The subroutine stephybrid advances the magnetic field and the electron pressure according to the hybrid equations by dt/(2*substeps). As time stepping algorithm a trapezoidal leapfrog scheme is used. One iteration consists of the following steps (only the first half of the leapfrog time step is shown):
1. the quantity on the right hand side of the evolution equation for B' is calculated (compare the hybrid 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/(2*substeps) using dB'/dt = E'
4. B at t+dt/(2*substeps) is calculated by inverting the Helmholtz equation  (1-d_e² div grad) B = B', call to subroutine helmholtz (source file multigrid.M)
5. step the electron pressure (unless the switch isothermal is used)
6. apply fourth order hyper viscosities to the magnetic field and the electron 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.
• Subroutine calc_e:

The subroutine calc_e calculates the electric field that enters the ion equation of motion (compare the hybrid equations): E = (j_e x B - grad p_e)/n
• Subroutine add_base_density:

Since in the hybrid equations the density appears at several location in the denominator numerical problems arise if the ion particle density becomes too small. To relax these problems an artificial homogeneous background density can be added with the routine add_base_density. The magnitude of the background density is defined by the parameter n_base.

• isothermal: If the switch #define isothermal is used the electron pressure is determined from the density and the constant electron temperature T_e.
• n_base: The parameter n_base sets the small artificial background density, for example #define n_base 0.1