5.2 Discretization

Both transient equations (5.2) and (5.6) are discretized with a forward Euler scheme, which guarantees reasonable stability.

Discretization of (5.2) leads to

$$E_1 = E_\mathrm{stat} + \tau_\mathrm{ef} \cdot \frac{E_1 - E_0}{t_1 - t_0}.$$ (5.7)

All the terms with the index 1 are related to the new time step, the ones with the index 0 to the old time step. By applying simple mathematics one gets

$$E_1 = \frac{E_\mathrm{stat} + \tau_\mathrm{ef} \cdot E_0}{1 + \tau_\mathrm{ef}}$$ (5.8)

for the explicit form. Similarly (5.6) reads as

$$P_1 = P_\mathrm{stat} - {\tau}_\mathrm{pol}\cdot \frac{P_1-P... ...n} \cdot (P - P_\mathrm{ef}) \cdot \frac{E_1 - E_0}{t_1 - t_0}$$ (5.9)

and the explicit equation as

$$P_1 = \frac{P_\mathrm{stat} + \mbox{\ c} \cdot P_0}{1 + \mbo... ..._h \cdot k_\mathrm{nonlin} \cdot \frac{E_1 - E_0}{t_1 - t_0}}.$$ (5.10)