A computational study is unlikely to lead to real scientific progress unless the software environment is convenient enough to encourage one to vary parameters, modify the problem, play around.
Numerical analysis is the study of the mathematics of the previous century.
If the answer is highly sensitive to perturbations, you have probably asked the wrong question. Scientific progress occurs by revolutions and paradigm shifts on all scales. If Kuhn had written The Structure of Scientific Revolutions after the popularization of fractals, he would never have dared to claim there were just two. Most scientists and engineers are in the continuous group, and most non-numerical computer scientists are in the discrete one. No physical constants are known to more than around eleven digits, and no truly scientific problem requires computation with much more precision than this.