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).