Thus far the emphasis in this book has been focused firmly on the solutions of IVPs and how well these are approximated by a variety of numerical methods. This attention is now shifted to the numerical method (primarily LMMs) and we ask whether the numerically computed values might be closer to the solution of a modified differential equation than they are to the solution of the original differential equation. At first sight this may appear to introduce an unnecessary level of complication, but we will see in this chapter (as well as those that follow on geometric integration) that constructing a new ODE that very accurately approximates the numerical method can provide important insights about our computations.