Lisl Weynans

''Classical'' Electropermeabilization Modeling at the Cell Scale

The aim of this paper is to provide new models of cell electropermeabilization involving only a few parameters. A static and a dynamical model, which are based on the description of the electric potential in a biological cell, are derived. Existence and uniqueness results are provided for each differential system, and an accurate numerical method to compute the solution is described. We then present numerical simulations that corroborate the experimental observations, providing the consistency of the modeling. We emphasize that our new models involve very few parameters, compared with the most achieved models of Neu and Krassowska (Phys Rev E 53(3):3471–3482, 1999) and DeBruin and Krassowska (Biophys J 77:1225–1233, 1999), but they provide the same qualitative results. Thus, these models will facilitate drastically the forthcoming inverse problem solving, which will consist in fitting them with the experiments.

A simple second order cartesian scheme for compressible Euler flows

We present a finite-volume scheme for compressible Euler flows where the grid is cartesian and it does not fit to the body. The scheme, based on the definition of an ad hoc Riemann problem at solid boundaries, is simple to implement and it is formally second order accurate. Error convergence rates with respect to several exact test cases are investigated and examples of flow solutions in one, two and three dimensions are presented.

A second-order Cartesian method for the simulation of electropermeabilization cell models

In this paper, we present a new finite differences method to simulate electropermeabilization models, like the model of Neu and Krassowska or the recent model of Kavian et al. These models are based on the evolution of the electric potential in a cell embedded in a conducting medium. The main feature lies in the transmission of the voltage potential across the cell membrane: the jump of the potential is proportional to the normal flux thanks to the well-known Kirchoff law. An adapted scheme is thus necessary to accurately simulate the voltage potential in the whole cell, notably at the membrane separating the cell from the outer medium. We present a second-order finite differences scheme in the spirit of the method introduced by Cisternino and Weynans for elliptic problems with immersed interfaces. This is a Cartesian grid method based on the accurate discretization of the fluxes at the interface, through the use of additional interface unknowns. The main novelty of our present work lies in the fact that the jump of the potential is proportional to the flux, and therefore is not explicitly known. The original use of interface unknowns makes it possible to discretize the transmission conditions with enough accuracy to obtain a second-order spatial convergence. We prove the second-order spatial convergence in the stationary linear one-dimensional case, and the first-order temporal convergence for the dynamical non-linear model in one dimension. We then perform numerical experiments in two dimensions that corroborate these results.

Super-convergence in maximum norm of the gradient for the Shortley-Weller method

We prove in this paper the second-order super-convergence in

A parallel second order Cartesian method for elliptic interface problems

L^{\infty }

Spray impingement on a wall in the context of the upper airways

L∞-norm of the gradient for the Shortley–Weller method. Indeed, this method is known to be second-order accurate for the solution itself and for the discrete gradient, although its consistency error near the boundary is only first-order. We present a proof in the finite-difference spirit, using a discrete maximum principle to obtain estimates on the coefficients of the inverse matrix. The proof is based on a discrete Poisson equation for the discrete gradient, with second-order accurate Dirichlet boundary conditions. The advantage of this finite-difference approach is that it can provide pointwise convergence results depending on the local consistency error and the location on the computational domain.