Rate of Convergence for a Nonlocal-to-local Limit in One Dimension

TL;DR

This paper quantifies the convergence rate of a nonlocal approximation to the local quadratic porous medium equation in one dimension, using Wasserstein distance and simple variational inequality techniques.

Contribution

It provides a novel, simple proof of convergence rate in 1D for a specific kernel, with numerical simulations indicating potential for improved rates.

Findings

Quantified convergence rate in 2-Wasserstein distance

Proof exploits Evolutionary Variational Inequality and a priori estimates

Numerical simulations suggest the rate can be improved

Abstract

We consider a nonlocal approximation of the quadratic porous medium equation where the pressure is given by a convolution with a mollification kernel. It is known that when the kernel concentrates around the origin, the nonlocal equation converges to the local one. In one spatial dimension, for a particular choice of the kernel, and under mere assumptions on the initial condition, we quantify the rate of convergence in the 2-Wasserstein distance. Our proof is very simple, exploiting the so-called Evolutionary Variational Inequality for both the nonlocal and local equations as well as a priori estimates. We also present numerical simulations using the finite volume method, which suggests that the obtained rate can be improved - this will be addressed in a forthcoming work.