We will present our novel approach to simulating the stochastic relativistic advection-diffusion equation using the Metropolis algorithm. We demonstrate that the dissipative dynamics of a boosted fluctuating fluid can be effectively simulated by performing random transfers of charge between fluid cells, combined with ideal hydrodynamic time steps. These random charge transfers are subjected to a Metropolis acceptance-rejection step, using entropy as the statistical weight. Our method successfully reproduces the expected behavior of dissipative relativistic hydrodynamics within a specific hydrodynamic frame, known as the density frame. We will share numerical results from our simulations, both with and without noise, and compare them to relativistic kinetics and analytical predictions. Uniquely, our approach to relativistic dissipative fluids is strictly first-order in gradients and does not introduce non-hydrodynamic modes, setting it apart from other numerical methods. Additionally, we will explore the (ongoing) extension of this formalism to relativistic viscous hydrodynamics and provide a comparative analysis with the BDNK formalism.