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

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 kappa-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 kappa-schemes. A further improvement is obtained by using so-called Corner Transport Upwind extension in two-dimensional case. The extended semi-implicit kappa-scheme with a specific (velocity dependent) value of kappa 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.Comment: arXiv admin note: substantial text overlap with arXiv:1611.04153 Comment from the authors - this is a corrected paper where typesetting error in formula (37) is remove

M.: A new level set method for motion in normal direction based on a semi-implicit forward-backward diffusion approach

Abstract. We introduce a new level set method for motion in normal direction. It is based on a formulation in the form of a second order forward-backward diffusion equation. The equation is discretized by the finite volume method. We propose a semi-implicit time discretization taking into account the forward diffusion part of the solution in an implicit way, while the backward diffusion part is treated explicitly. When forward diffusion dominates, a straightforward reconstruction of the solution is used, while larger (smoothing) stencils are used when backward diffusion dominates. The method is precise on coarse grids and is second order accurate for smooth solutions. Numerical experiments show an optimal coupling of time and space steps with τ = h, and no stronger CFL condition is required. Numerical tests with the scheme are discussed on representative examples