Definitions
Absolute stability
Forward Euler vs. Backward Euler
The Curtiss-Hirschfelder example
The test problem [Curtiss & Hirschfelder, 1952]
(1)with different initial conditions $y(0) = y_0$ is a classical example of a stiff system. Indeed, it exhibits two widely different time scales, one corresponding to the slow manifold $y^S(t) = \cos(t)$ and the strongly damped mode $e^{-50t}$, as illustrated by the solution curves (light gray) in the figures below.
The absolute stability requirement for the FE method applied to this problem leads to the stepsize restriction $h \le 2/50$. The figure below shows the results of integrating this problem with the FE method and a fixed stepsize $h=2.01/50$.
Using a stepsize within the region of absolute stability, leads to a stable method (note that, for accuracy requirements, one may require a smaller stepsize)
On the other hand, the BE method is stable even with a stepsize $h=0.5$, as shown in the following figure (a CVODE solution with relative and absolute tolerances of $10^{-4}$ is also shown).
BDF and Adams-Moulton regions of absolute stability
The absolute stability regions for BDF of orders 1 to 6 are shown in the figure below. Note that the stability regions are outside the shaded area for each method.
The absolute stability regions for Adams-Moulton methods of orders 1-9 are shown in the figure below. Note that the first two methods — backward Euler (p=1) and trapezoidal (p=2) — are A-stable.
