We discuss an initial value problem for an implicit second order ordinary differential equation which arises in models of flow in saturated porous media such as concrete. Depending on the initial condition, the solution features a sharp interface with derivatives which become numerically unbounded. By using an integrator based on finite difference methods and equipped with adaptive step size selection, it is possible to compute the solution on highly irregular meshes. In this way it is possible to verify and predict asymptotical theory near the interface with remarkable accuracy.
Amodio, P., Budd, C. J., Koch, O., Settanni, G., & Weinmüller, E. (2014). Asymptotical computations for a model of flow in saturated porous media. Applied Mathematics and Computation, 237, 155-167. https://doi.org/10.1016/j.amc.2014.03.063