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 2 nd 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 3 rd 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.

A new numerical method for level set motion in normal direction used in optical flow estimation

“…In what follows we assume that |∇f n ij | = 0, and we define "directional Courant numbers" [10,6] for the fixed time step τ ,…”

“…In what follows, we modify the scheme (15) using an idea of so called Corner Transport Upwind scheme [4,10,6]. Opposite to the scheme (15) that uses the linear interpolation of three values, the following scheme will use the bilinear interpolation involving also the corner value f n i+kj+l , namely…”

“…For that purpose, we extend a so-called Rouy-Tourin scheme [13,12] for the numerical solution of nonlinear advection equation using an idea of the so-called Corner Transport Upwind scheme that is described for linear advection equation in [4,10,6]. The Rouy-Tourin scheme is based on linear interpolation, while the proposed scheme uses the bilinear interpolation.…”

Semi-implicit methods for advection equations with explicit forms of numerical solution

“…Implicit (or semi-implicit) numerical schemes are useful numerical methods to solve advection dominated problems in several circumstances [1,2,3,4,5,6,7,8,10,14,15,17,18,19,20,22]. They can avoid or reduce significantly the main disadvantage of fully explicit schemes that are implemented on a fixed mesh with a finite stencil in numerical discretization.…”

Finite element method for two-dimensional linear advection equations based on spline method

“…This approach is also used in solving the linear advection equations which have a time dependent domain given by positions of a fire front in a forest [7]. A new representation of semi-implicit scheme is constructed in [8], a novel approach of partial Lax-Wendroff procedure is used in this new scheme.…”