Numerical boundary conditions for ODE

We consider consistent finite difference approximations of ordinary differential equations, and in particular, parasitic solutions. A framework is introduced, representing a discrete solution as a sum of the true solution and a number of parasitic solutions. We show that within this framework, finite difference equations can be analysed using theory of ordinary differential equations, simplifying the analysis considerably. As an example we give a simple recipe on how to construct numerical boundary conditions such that the solution converges with expected accuracy