Garrett Birkhoff and Richard S. Varga, Discretization errors for well-set Cauchy problems. I, J. Math. and Phys. 44 (1965), 1-23. MR 31, 4189. Zbl. 134, p. 134.

1. Introduction. It is generally believed that, given a well-set Cauchy
problem, there exist accurate and numerically "stable" *difference
approximations* to the differential equations (DE's) defining the problem,
from which (in principle) the solution can be computed to any desired degree of
accuracy in finite time. However, to do this in practice is a very different
matter.

Specifically, one must construct a difference approximation which:

(i) simulates the DE's to a reasonably high *order of accuracy*, and

(ii) is *numerically stable* to small errors (e.g., of roundoff).