Windowing errors

Boundary conditions have to be included by careful treatment of the discretised differential operator near the boundaries. This introduces no error if the spatial windows are finite. If the real spatial window is infinite, then the approximation of using a finite spatial window introduces further errors. This can be checked through doubling the window-size, while keeping the step-sizes constant. Note that using mapping techniques also reduces windowing errors. However, mapping does not really eliminate them totally. Instead, it transforms windowing errors into new spatial-discretisation errors--caused by the rapidly increasing spatial variation of the equation terms, as the transformed boundaries in the mapping space are approached. Even so, this method illustrates how mapping techniques can be combined with a numerical method.