13 Finite Difference Methods
Most problems met in practice have no formula and no series; they are solved on a computer. The oldest and most direct method replaces derivatives by difference quotients on a grid. Richardson tried in 1922 to forecast the weather this way by hand, and his forecast failed, not because of the arithmetic but because his scheme was unstable. The theory that explains such failures, due to Courant, Friedrichs and Lewy (1928) and to von Neumann in the 1940s, is the subject of this section. Its central lesson is that a scheme which approximates the equation well may still produce nonsense, and that stability is a property to be checked, not assumed.
13.1 Difference quotients and consistency
13.2 Stability: the von Neumann method
13.3 Implicit schemes
13.4 Hyperbolic equations and the CFL condition
13.5 Finite differences in software
13.6 Practice Problems
13.2 Stability: the von Neumann method
13.3 Implicit schemes
13.4 Hyperbolic equations and the CFL condition
13.5 Finite differences in software
13.6 Practice Problems
Questions on this section
Stuck on something here? Ask below and it stays attached to this topic.