Trapezoidal rule fails B-stability
= Trapezoidal rule fails B-stability
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 $y=\sqrt2$, $T=0$ and implicit differentiation gives $T'=-2$, so nearby states expand rather than contract. This is a concrete nonlinear obstruction, rather than merely failure of the sufficient <algebraic stability> criterion.