Isotropic finite-difference discretization of stochastic conservation laws preserving detailed balance

The dynamics of thermally fluctuating conserved order parameters are described by stochastic conservation laws. Thermal equilibrium in such systems requires the dissipative and stochastic components of the flux to be related by detailed balance. Preserving this relation in spatial and temporal discretization is necessary to obtain solutions that have fidelity to the continuum. Here, we propose a finite-difference discretization that preserves detailed balance on the lattice, has spatial error that is isotropic to leading order in lattice spacing, and can be integrated accurately in time using a delayed difference method. We benchmark the method for model B dynamics with a $\phi^{4}$ Landau free energy and obtain excellent agreement with analytical results.