Global Error Propagation in the Numerical Solution of an Ordinary Differential Equation with a Discontinuity

We consider the numerical solution of a ordinary differential equation with a discontinuity in its derivative. Specifically, we assume that the location of the discontinuity has been detected, but that a numerical error exists in this location. We study how this error propagates in the subsequent numerical solution beyond the discontinuity. We derive an analytical expression for this error, which exposes terms that can only be controlled, if possible, prior to the advent of the discontinuity. We also find a condition number that can amplify the location error significantly. An example illustrates our findings.