From noise on the sites to noise on the links: discretizing the conserved Kardar-Parisi-Zhang equation in real space

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.
Comment: Multiplicative noise case added, 6 figures