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

TL;DR

This paper introduces a stochastic particle algorithm to approximate solutions of a one-dimensional generalized porous media PDE, extending previous stochastic representations to irregular coefficients and comparing with deterministic methods.

Contribution

It develops a novel stochastic particle algorithm for solving porous media type PDEs with possibly irregular coefficients, extending existing stochastic representations.

Findings

Algorithm effectively approximates PDE solutions.

Supports irregular coefficient functions.

Provides comparison with deterministic numerical methods.

Abstract

The object of this paper is a one-dimensional generalized porous media equation (PDE) with possibly discontinuous coefficient $\beta$, which is well-posed as an evolution problem in $L^1(\mathbb{R})$. In some recent papers of Blanchard et alia and Barbu et alia, 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 $\beta$ is continuous and the initial condition is only integrable with some supplementary technical assumption. The main purpose of the article consists in introducing and implementing a stochastic particle algorithm to approach the solution to (PDE) which also fits in the case when $\beta$ is possibly irregular, to predict some long-time behavior of the solution and in comparing with some recent numerical deterministic techniques.