A real <vector field> $f$ is dissipative in an <inner product> if
$$
\langle f(x)-f(y),x-y\rangle\leq0
$$
for all $x,y$. Two solutions of the <ordinary differential equation> $y'=f(y)$ satisfy
$$
\frac{d}{dt}\|x(t)-y(t)\|^2
=2\langle f(x(t))-f(y(t)),x(t)-y(t)\rangle\leq0.
$$
Thus their distance never increases. The linear case $f(y)=Ay$ recovers the <dissipative operator> condition. <B-stability> asks whether a <Runge-Kutta method> preserves this contraction for every positive <step size>.


 Dissipative vector field

ID: dissipative-vector-field