A Local Semi-Implicit Level-Set Method for Interface Motion

Abstract

This paper proposes and implements a novel hybrid level set method which combines the numerical efficiency of the local level set approach with the temporal stability afforded by a semi-implicit technique. By introducing an extraction/insertion algorithm into the local level set approach, we can accurately capture complicated behaviors such as interface separation and coalescence. This technique solves a well known problem when treating a semi-implicit system with spectral methods, where spurious interface movements emerge when two interfaces are close to each other. Numerical experiments show that the proposed method is stable, efficient and scales up well into three dimensional problems.

A Semi-implicit Gradient Augmented Level Set Method

“…To determine a smooth velocity field elsewhere in the domain values can be extended from the interface through the use of an extension equation applied to a quantity q [14,15,17],…”

“…(2.19) or (2.20) at grid points next to the interface. Extend κ by at least 4 grid points using an extension algorithm [14,15,17]. if velocity field is surface diffusion then Set S = 0 in the entire domain.…”

“…A fully implicit method would alleviate the time step restriction but would add the complexity of solving a possibly non-linear system every time step. Another approach is to extract the linear portion of the advection equation and treat it implicitly, with the non-linear part treated explicitly [14,15]. This has the advantage of numerically stabilizing the system at the cost of a slight deterioration of the accuracy of the solution.…”

“…The original gradient augmented level set method was demonstrated using analytic flow fields and demonstrated excellent accuracy with minimal additional computational cost. The use of the augmented gradient level set method in modelling vesicle motion [7] has shown that while the time step restriction is not as stringent as regular explicit schemes it is not possible to take time steps on the order of the grid spacing, as is possible with semi-implicit methods [14,15].…”

The Explicit–Implicit–Null method: Removing the numerical instability of PDEs

“…In [5], we presented a rough sketch of the idea, and used it to stabilize the viscous free-surface dynamics of two liquid drops during coalescence. Subsequently, similar ideas have been implemented to stabilize the motion of a surface in the diffuse interface and level-set methods [6,7,8], and for the solution of PDEs on surfaces [9].…”