for , subject to the mixed spatial boundary conditions

(192) |

(193) |

Equation (191) needs to be discretized in both time and space.
In time, we discretize
on the equally spaced grid

(194) |

In space, we discretize on the usual equally spaced grid-points specified in Eq. (114), and approximate via the second-order, central difference scheme introduced in Eq. (115). The spatial boundary conditions are discretized in a similar manner to Eqs. (134) and (135). Thus, Eq. (195) yields

(196) |

for , where , and . The discretized boundary conditions take the form

where ,

Equations (197)-(199) constitute a fairly straightforward iterative scheme which can be used to evolve the in time.