In this paper, we provide a scheme for simulating one-dimensional processes generated by divergence or non-divergence form operators with discontinuous coefficients. We use a space bijection to transform such a process in another one that behaves locally like a Skew Brownian motion. Indeed the behavior of the Skew Brownian motion can easily be approached by an asymmetric random walk.
"On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients." Electron. J. Probab. 11 249 - 275, 2006. https://doi.org/10.1214/EJP.v11-311