In one dimension, the 2-Wasserstein distance between the nonlocal and local porous medium solutions is bounded by a constant times the square root of the smoothing parameter.
A deterministic particle method for the porous media equation
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This paper deals with the deterministic particle method for the equation of porous media (with p = 2). We establish a convergence rate in the Wasserstein-2 distance between the approximate solution of the associated nonlinear transport equation and the solution of the original one. This seems to be the first quantitative rate for diffusion-velocity particle methods solving diffusive equations and is achieved using a novel commutator estimate for the Wasserstein transport map.
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Rate of Convergence for a Nonlocal-to-local Limit in One Dimension
In one dimension, the 2-Wasserstein distance between the nonlocal and local porous medium solutions is bounded by a constant times the square root of the smoothing parameter.