ABSTRACT
The classical analysis of numerical methods for time-dependent, ordinary or partial differential equations is based on the ideas of stability, consistency and convergence. Roughly speaking, consistency means small local errors and stability means that errors do not build up catastrophically. Together, consistency and stability yield convergence: small (global) errors. However it is clear that there are useful theoretical properties of a method beyond its consistency, stability and convergence. Here we are interested in conserved quantities: the differential equations being integrated may possess one or several quantities (mass, energy, etc.) that are conserved in the true evolution and it is reasonable to demand that the numerical scheme also preserves those quantities. Several reasons are usually invoked for using schemes with such conservation properties. In a recent paper [6], C.W. Gear writes “In some cases, failure to maintain certain invariants leads to physically impossible solutions”. In other cases conservation quantities are deemed important to avoid spurious blow-up of the numerical solution. In a classical paper [1], Arakawa writes “If we can find a finite difference scheme which has constraints analogous to the integral constraints of the differential form, the solution will not show the false ‘noodling’, followed by computational instability”.
