Trapezoidal rule

The trapezoidal ODE rule averages vector fields at the old and new states and is A-stable.

The trapezoidal rule is A-stable but is not B-stable. For the dissipative vector field $f(y)=-y^3$ at step size one, its uniquely defined step map $T$ obeys $T+T^3/2=y-y^3/2$. At $y=\sqrt{2}$, $T=0$ and implicit differentiation gives $T^{\prime }=-2$, so nearby states expand rather than contract. This is a concrete nonlinear obstruction, rather than merely failure of the sufficient algebraic stability criterion.