  Numerical analysis of conserved field dynamics has been generally performed
with pseudo spectral methods. Finite differences integration, the common
procedure for non-conserved field dynamics, indeed struggles to implement a
conservative noise in the discrete spatial domain. In this work, we present a
novel method to generate a conservative noise in the finite differences
framework, which works for any discrete topology and boundary conditions. We
apply it to numerically solve the conserved Kardar-Parisi-Zhang (cKPZ)
equation, widely used to describe surface roughening when the number of
particles is conserved. Our numerical simulations recover the correct scaling
exponents $\alpha$, $\beta$, and $z$ in $d=1$ and in $d=2$. To illustrate the
potentiality of the method, we further consider the cKPZ equation on different
kinds of non-standard lattices and on the random Euclidean graph. This is the
first numerical study of conserved field dynamics on an irregular topology,
paving the way to a broad spectrum of possible applications.
