Fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and Newton methods

This framework is concerned with the numerical modeling of the dynamics of isolated biomembranes and capillary interfaces in a surrounding Newtonian fluid. A level set approach helps to follow the interface motion. Our method features the use of high order fully implicit time integration schemes that enable to overcome stability issues related to the explicit discretization of the highly non-linear bending force or capillary force. At each time step, the tangent systems are derived and the resulting nonlinear problems are solved by a Newton-Raphson method. Based on the signed distance assumption, several inexact Newton strategies are employed to solve the capillary and vesicle problems and guarantee the second-order convergence behavior. We address in detail the main features of the proposed method, and we report several experiments in the two-dimensional case with the aim of illustrating its accuracy and efficiency. Comparative investigations with respect to the fully explicit scheme depict the stabilizing effect of the new method, which allows to use significantly larger time step sizes.