Karol Mikula

Semi-implicit Level Set Method with Inflow-Based Gradient in a Polyhedron Mesh

In this paper , a semi-implicit method is proposed to solve a propagation in a normal direction with a cell-centered finite volume method. An inflow-based gradient is used to discretize the magnitude of the gradient and it brings the second order upwind difference in an evenly spaced one dimensional domain. In three dimensional domain, we numerically verify that the proposed scheme is second order. The implementation is straightforwardly combined with a conventional finite volume code and 1-ring face neighborhood for parallel computation. An experimental order of convergence and a comparison of wall clock time between semi-implicit and explicit method are illustrated by numerical examples.

Semi-implicit second order schemes for numerical solution of level set advection equation on Cartesian grids

Abstract A new parametric class of semi-implicit numerical schemes for a level set advection equation on Cartesian grids is derived and analyzed. An accuracy and a stability study is provided for a linear advection equation with a variable velocity using partial Lax–Wendroff procedure and numerical von Neumann stability analysis. The obtained semi-implicit κ -scheme is 2nd order accurate in space and time in any dimensional case when using a dimension by dimension extension of the one-dimensional scheme that is not the case for analogous fully explicit or fully implicit κ -schemes. A further improvement is obtained by using so-called Corner Transport Upwind extension in two-dimensional case. The extended semi-implicit κ -scheme with a specific (velocity dependent) value of κ is 3rd order accurate in space and time for a constant advection velocity, and it is unconditional stable according to the numerical von Neumann stability analysis for the linear advection equation in general.

Semi-implicit Second Order Accurate Finite Volume Method for Advection-Diffusion Level Set Equation

We present a second order accurate finite volume method for level set equation describing the motion in normal direction with the speed depending on external properties and curvature. A convenient combination of a Crank-Nicolson type of the time discretization for diffusion term [1] and an Inflow Implicit and Outflow Explicit scheme [6] for advection term is used. Numerical experiments for an example with the exact solution derived in this paper and for examples motivated by modeling of fire propagation in forests are presented.

3D Lagrangian Segmentation with Simultaneous Mesh Adjustment

We present a method for 3D image segmentation based on the Lagrangian approach. The segmentation model is a 3D analogue of the geodesic active contour model [1] and it contains an additional tangential movement term that allows us to control the quality of the mesh during the evolution process. The model is discretized by the finite volume approach. Segmentation of zebrafish cell images is shown to illustrate the performance of the method.