A probabilistic algorithm approximating solutions of a singular PDE of porous media type

The object of this paper is a one-dimensional generalized porous media equation (PDE) with possibly discontinuous coefficient β , which is well-posed as an evolution problem in L ¹(R) In some recent papers of Blanchard et al. and Barbu et al., the solution was represented by the solution of a non-linear stochastic differential equation in law if the initial condition is a bounded integrable function. We first extend this result, at least when β is continuous and the initial condition is only integrable with some supplementary technical assumption. The main purpose of the article is to introduce and implement a stochastic particle algorithm to approach the solution to PDE, which also fits in the case when β is irregular. We compare it with some recent numerical deterministic techniques. As a byproduct, we can predict the long-time behavior of the solution in several cases.