Unconditionally stable higher order semi-implicit level set method for advection equations

TL;DR

This paper introduces unconditionally stable, high-order semi-implicit finite difference schemes for advection equations in level set methods, enabling accurate and efficient simulations even at large Courant numbers.

Contribution

It develops a third-order accurate, unconditionally stable semi-implicit scheme for linear advection with space-dependent velocity, and proposes a TVD scheme for spatial derivatives, improving stability and accuracy.

Findings

Third-order accuracy for linear advection with variable velocity.

Unconditional stability confirmed by von Neumann analysis.

Efficient convergence with large Courant numbers.

Abstract

We present compact semi-implicit finite difference schemes on structured grids for numerical solutions of the advection by an external velocity and by a speed in normal direction that are applicable in level set methods. The most involved numerical scheme is third order accurate for the linear advection with a space dependent velocity and unconditionally stable in the sense of von Neumann stability analysis. We also present a simple high-resolution scheme that gives a TVD (Total Variation Diminishing) approximation of the spatial derivative for the advected level set function. In the case of nonlinear advection, the semi-implicit discretization is proposed to linearize the problem. The compact form of implicit stencil in numerical schemes containing unknowns only in the upwind direction allows applications of efficient algebraic solvers like fast sweeping methods. Numerical tests to evolve a smooth and non-smooth interface and an example with a large variation of velocity confirm the good accuracy of the methods and fast convergence of the algebraic solver even in the case of very large Courant numbers.