Communications in Mathematical Sciences

Volume 7 (2009)

Number 4

A diffuse-interface approach for modelling transport, diffusion and adsorption/desorption of material quantities on a deformable interface

Pages: 1009 – 1037



Xiangrong Li

John Lowengrub

Knut Erik Teigen

Axel Voigt

Fan Wang


A method is presented to solve two-phase problems involving a material quantity on an interface. The interface can be advected, stretched, and change topology, and material can be adsorbed to or desorbed from it. The method is based on the use of a diffuse interface framework, which allows a simple implementation using standard finite-difference or finite-element techniques. Here, finite-difference methods on a block-structured adaptive grid are used, and the resulting equations are solved using a non-linear multigrid method. Interfacial flow with soluble surfactants is used as an example of the application of the method, and several test cases are presented demonstrating its accuracy and convergence.


Partial differential equations, diffuse interface, interfacial dynamics, complex geometry, multigrid, adaptive grid, finite difference, multiphase, adsorption, desorption

2010 Mathematics Subject Classification

35K05, 35K57, 35Q35, 65M06, 65M50, 65M55, 65Z05, 76Txx, 82C24

Published 1 January 2009