4. Unsteady equations

In the previous sections, only the spatial direction was discretized. Unsteady problems involve an additional variable: time t, which describes a bounded interval [0, T], T > 0, and has a special status. The spectral discretization of unsteady problems mostly respects this difference in status; a temporal discretization scheme is used, based on an e.g. regular slicing of [0, T] into intervals [t _n , t _{n + 1} ], with a constant time step δt = t _{n + 1} - t _n . The solution of the unsteady problems u (x, t _n ) at time t _n is then approximated by a quantity u _n (x), recursively...