Unsteady equations
Discovering spectral methods

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 defined by a scheme, of implicit or explicit type: this scheme consists in solving, at each time step t_n a partial differential equation in space. Each of these problems must then be discretized by an approximation in space, which may be one of the...