Skip to Main content Skip to Navigation
Journal articles

Superconvergent second order Cartesian method for solving free boundary problem for invadopodia formation

Olivier Gallinato 1, 2, 3 Clair Poignard 3, 2 
3 MONC - Modélisation Mathématique pour l'Oncologie
IMB - Institut de Mathématiques de Bordeaux, Institut Bergonié [Bordeaux], Inria Bordeaux - Sud-Ouest
Abstract : In this paper, we present a superconvergent second order Cartesian method to solve a free boundary problem with two harmonic phases coupled through the moving interface. The model recently proposed by the authors and colleagues describes the formation of cell protrusions. The moving interface is described by a level set function and is advected at the velocity given by the gradient of the inner phase. The finite differences method proposed in this paper consists of a new stabilized ghost fluid method and second order discretizations for the Laplace operator with the boundary conditions (Dirichlet, Neumann or Robin conditions). Interestingly, the method to solve the harmonic subproblems is superconvergent on two levels, in the sense that the first and second order derivatives of the numerical solutions are obtained with the second order of accuracy, similarly to the solution itself. We exhibit numerical criteria on the data accuracy to get such properties and numerical simulations corroborate these criteria. In addition to these properties, we propose an appropriate extension of the velocity of the level-set to avoid any loss of consistency, and to obtain the second order of accuracy of the complete free boundary problem. Interestingly, we highlight the transmission of the superconvergent properties for the static subproblems and their preservation by the dynamical scheme. Our method is also well suited for quasistatic Hele-Shaw-like or Muskat-like problems.
Document type :
Journal articles
Complete list of metadata
Contributor : Clair Poignard Connect in order to contact the contributor
Submitted on : Monday, March 6, 2017 - 2:58:35 AM
Last modification on : Wednesday, February 2, 2022 - 3:54:13 PM
Long-term archiving on: : Wednesday, June 7, 2017 - 12:26:07 PM


Files produced by the author(s)


Distributed under a Creative Commons Attribution 4.0 International License



Olivier Gallinato, Clair Poignard. Superconvergent second order Cartesian method for solving free boundary problem for invadopodia formation. Journal of Computational Physics, Elsevier, 2017, 339, pp.412 - 431. ⟨10.1016/⟩. ⟨hal-01483484⟩



Record views


Files downloads