  We study an approach to simulating the stochastic relativistic
advection-diffusion equation based on the Metropolis algorithm. We show that
the dissipative dynamics of the boosted fluctuating fluid can be simulated by
making random transfers of charge between fluid cells, interspersed with ideal
hydrodynamic time steps. The random charge transfers are accepted or rejected
in a Metropolis step using the entropy as a statistical weight. This procedure
reproduces the expected strains of dissipative relativistic hydrodynamics in a
specific (and non-covariant) hydrodynamic frame known as the density frame.
Numerical results, both with and without noise, are presented and compared to
relativistic kinetics and analytical expectations. An all order resummation of
the density frame gradient expansion reproduces the covariant dynamics in a
specific model. In contrast to all other numerical approaches to relativistic
dissipative fluids, the dissipative fluid formalism presented here is strictly
first order in gradients and has no non-hydrodynamic modes. The physical
naturalness and simplicity of the Metropolis algorithm, together with its
convergence properties, make it a promising tool for simulating stochastic
relativistic fluids in heavy ion collisions and for critical phenomena in the
relativistic domain.
