The error is monitored during the calculation and the grid structure is modified as needed. The number of grid points required can be reduced by orders of magnitude in three space dimensions and so, presumably, can the required number of observations. Any numerical method must be stable to be of practical use. All stability concepts give a bound for how fast errors in the numerical solution will grow. The errors may come from different sources, such as measurement errors, round-off errors introduced by floating point arithmetic, etc.
Stability is arguably the most important numerical property.
Consequently, many different kinds of stability concepts have been developed. It can be applied to most problems that arise in physics and engineering. It handles geometrically complex problems very well. The original PDEs and their numerical approximations can be analyzed by applying identical principles in the two cases, which simplifies the design of the numerical scheme. A systematic stability theory has been developed for high-order finite difference approximations of hyperbolic and parabolic PDE systems.