It seems that interval arithmetic can be used to quantify floating point truncation error in computational calculations. Does anyone know if it is possible to use interval arithmetic in a finite difference scheme, to automatically quantify the error of the scheme?
For example, using interval arithmetic, given some complicated f(x), and given a finite precision interval which contains the infinite precision value of x, I can get an exact representation (in finite precision) of an interval which contains f(x). This will allow me to know the impact of the truncation/rounding errors that have occurred during the calculation.
Now, if f(N,x) is an N-point approximation to f(x), can I somehow use interval arithmetic to determine a good interval [f1, f2] which will contain f(x) when computed using the approximation f(N,x)?
In particular, I'm wondering if I can get an error bound of f(x) when it is the solution to a PDE which is solved using a finite difference scheme.
While I have seen this alluded to, I have not seen a good description or reference.