Operator splitting and multigrid methods

In general we will be searching for methods of solving problems containing terms where the courant condition is too restrictive for practical computation; typical in this regard is the inclusion of diffusion or thermal conduction which contain second derivatives in space implying that time steps must scale as ∆t ∝ 1/(∆x). In this case we can not solve the problems explicitly but rather solve (at least part of) the problem implicitly to avoid the time step restrictions imposed by the inclusion of these operators. We follow Numerical Recipes and consider an initial value problem of the form