  We study scaling limits of the weakly driven Zhang and the
Bak-Tang-Wiesenfeld (BTW) model for self-organized criticality. We show that
the weakly driven Zhang model converges to a stochastic partial differential
equation (PDE) with singular-degenerate diffusion. In addition, the
deterministic BTW model is shown to converge to a singular-degenerate PDE.
Alternatively, the proof of the scaling limit can be understood as a
convergence proof of a finite-difference discretization for singular-degenerate
stochastic PDEs. This extends recent work on finite difference approximation of
(deterministic) quasilinear diffusion equations to discontinuous diffusion
coefficients and stochastic PDEs. In addition, we perform numerical simulations
illustrating key features of the considered models and the convergence to
stochastic PDEs in spatial dimension $d=1,2$.
