A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation

Fausto Cavalli; Giovanni Naldi

doi:10.3934/krm.2010.3.123

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2010, Volume 3, Issue 1: 123-142. Doi: 10.3934/krm.2010.3.123

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A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation

Fausto Cavalli^1, and
Giovanni Naldi^2,

1.
Università degli Studi di Brescia, Dipartimento di Matematica, Via Valotti 9, 25133 Brescia
2.
Università degli Studi di Milano, Dipartimento di Matematica "F. Enriques”, Via Saldini 50, 20133 Milano

Received: October 31, 2009

Revised: November 30, 2009

Published: February 2010

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Abstract

In this work we introduce a new numerical approach for solving Cahn-Hilliard equation with Neumann boundary conditions involving recent mass transportation methods. The numerical scheme is based on an alternative formulation of the problem using the so called pseudo-inverse of the cumulative distribution function. We establish a stable fully discrete scheme that inherits the energy dissipation and mass conservation from the associated continuous problem. We perform some numerical experiments which confirm our results.

Keywords:

Cahn-Hilliard equation; pseudo-inverse function; stable numerical methods for fourth order equations.

Mathematics Subject Classification: Primary: 65M06, 65M12; Secondary: 35Q70.

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A Wasserstein approach to the numerical solution of the one-dimensional Cahn-Hilliard equation

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