  The renormalization constants present in the lattice evaluation of the
topological susceptibility can be non-perturbatively calculated by using the
so-called heating method. We test this method for the $O(3)$ non-linear
$\sigma$-model in two dimensions. We work in a regime where perturbative
calculations are exact and useful to check the values obtained from the heating
method. The result of the test is positive and it clarifies some features
concerning the method. Our procedure also allows a rather accurate
determination of the first perturbative coefficients.
