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

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.

Tumor growth model of ductal carcinoma: from in situ phase to stroma invasion

“…The difficulty lies in the method used to tackle the ellipic problem satisfied by the pressure (11d)-(11e). The numerical method is inspired from recent works of O. Gallinato and C. Poignard [12,13], in which the elliptic operator and the Neumann boundary condition are discretized thanks to a stabilized version of the usual Ghost fluid method [8], based on the continuity of the stencils. Standard upwind schemes are considered to compute the advectionreaction equations, and forward Euler time-schemes are used for the time discretization.…”

“…at the regular points. (13) Near the interface the centered discretization is not possible since one (at least) of the neighbors is on the other side of the interface. At this point, the value is called ghost value and is linearly extrapolated.…”

“…If the case θ x = 1 occurs, we have u Γ h = u h i−1 . The scheme (14) is then equivalent to the standard 3-point stencil scheme (13). Conversely, if θ x = 0, the point x i is an interface point and does not belong to the considered domain.…”

Solving elliptic interface problems with jump conditions on Cartesian grids

“…While this method produces symmetric positive definite linear systems and only requires the right-hand side of the linear system to be modified, it requires the generation of a local Voronoi mesh and interpolation of numerical solutions from such unstructured meshes back onto Cartesian grids, which may add some challenges, especially in three spatial dimensions. The literature on solving elliptic problems with jump conditions is quite vast and we refer the interested reader to the review [28] and to other approaches, such as cut-cell approaches [16,52], discontinuous Galerkin and the eXtended Finite Element Method (XFEM) [38,32,48,17,7,47,34,21,29,62], the Virtual Node Method [49,6,49,60,56,36] or other fictitious domain approaches [15,14,23].…”

A Pore-Scale Model for Permeable Biofilm: Numerical Simulations and Laboratory Experiments

“…Further, the biofilm-water interface location changes over time, and therefore is a free boundary problem. Numerical methods for solving free boundary problems are an active research field ( [12,13]). The arbitrary Lagrangian-Eulerian (ALE) method is used to track the position of the biofilm-water interface ( [8]).…”