Real space approaches to solving partial differential equations have a number of appealing properties. The updates are local in space; the completely nonlocal Coulomb problem has been turned into one of multiple iterations, each one of which uses only information in the immediate neighborhood of a given grid point. If there is some physically local effect such as motion of a charge followed by balancing charge relaxation around the moved charge, one only needs iterate in that region of space. Also, when the potential field is solved for a given configuration of charges, that field can be used as the input for the next configuration, which presumably is often close to the previous one. Finally, with the advent of multigrid methods (discussed below), the electrostatic part of the problem can be solved in a way which scales linearly with system size. Disadvantages are that a very large number of grid points may be required, and highly accurate results may thus be difficult to obtain. One solution to this problem is to go to higher order difference equations for a relatively low extra cost in computing time per relaxation step, as discussed below (Sect. 2.4.4). Another problem is that it may be difficult to compute accurate forces numerically on the grid if dynamical information is desired.
If one uses a standard real space SOR relaxation algorithm on a given scale, invariably the solution process will stall after the short wavelength modes of the error have been removed. This led to the development of multiscale methods, discussed next.