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

TL;DR

This paper introduces a new class of semi-implicit second order schemes for solving the level set advection equation on Cartesian grids, offering improved accuracy and stability over existing explicit or implicit methods.

Contribution

The paper develops and analyzes a semi-implicit kappa-scheme that is second order accurate in multiple dimensions and can be extended to third order with Corner Transport Upwind, enhancing stability and accuracy.

Findings

The semi-implicit kappa-scheme is 2nd order accurate in space and time.

The scheme achieves 3rd order accuracy with Corner Transport Upwind extension.

It is unconditionally stable for linear advection equations.

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 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.