Fast Runge-Kutta approximation of inhomogeneous parabolic equations

The result after $N$ steps of an implicit Runge-Kutta time discretization of an inhomogeneous linear parabolic differential equation is computed, up to accuracy $\epsilon$, by solving only $$O\Big(\log N \log \frac1\epsilon \Big) $$ linear systems of equations. We derive, analyse, and numerically illustrate this fast algorithm.