Treating a single, stiff, second-order ODE directly (Q1262098)

A single autonomous linear, constant coefficient, second order differential equation is selected as a model problem to consider stability. Three direct methods based on interpolating back values of \(y^{(2-j)}\), \(j=0,1,2\) are studied. For these, stability regions are characterized using the two arbitrary coefficients of the model problem. The methods for \(j=1\) and 2 are appropriate for treating stiff problems. Based on a suitable implementation for a nonlinear problem, the authors propose a strategy for switching between methods to accomodate stiffness effectively. Two non-autonomous numerical examples illustrate this effectiveness. The novel characterization of stability, and the advantages to be realized through the proposed strategy make this article recommended reading.