Time Discretization

  We have given discretization formulae for the spatial differential operators which do not depend on time. The time derivative of (4.1-4) is related to each grid node and can be seen as a transient variation of the solution variable. The generation/recombination rate is also related to the grid nodes and is simply accounted for by addition or subtraction of the generated/recombinated species to the solution variables. As the equation system resulting from discretization is extremely stiff, i.e. the time constants can vary by orders of magnitude, implicit time discretization schemes are used. We apply the backward-Euler method for the time discretization (4.1-17), where gives the solution variable at the grid point i for the actual time step and the solution variable for the next time step . The backward-Euler scheme is unconditionally stable for any time step size [Vem81]. The time derivative is then calculated by (4.1-18), where the time step difference is referred to as time step size (4.1-19).