  We derive rigorous truncation-error bounds for the spin-boson model and its
generalizations to arbitrary quantum systems interacting with bosonic baths.
For the numerical simulation of such baths the truncation of both, the number
of modes and the local Hilbert-space dimensions is necessary. We derive
super-exponential Lieb--Robinson-type bounds on the error when restricting the
bath to finitely-many modes and show how the error introduced by truncating the
local Hilbert spaces may be efficiently monitored numerically. In this way we
give error bounds for approximating the infinite system by a finite-dimensional
one. As a consequence, numerical simulations such as the time-evolving density
with orthogonal polynomials algorithm (TEDOPA) now allow for the fully
certified treatment of the system-environment interaction.
