Conservative Level Set Method

where n is the normal of the interface, and ψ(1-ψ)n is the artificial compression flux which will act in the normal direction of the interface and in regions where 0<ψ<1. In this equation τ and ∇.(ϵ(∇ψ.n)n) are the artificial time and diffusion term respectively. This equation is advanced in pseudo-times τ. Compression term on the left-hand side aims at sharpening the profile, and diffusion term on the right-hand side will ensure the profile remains of the characteristic thickness ϵ. As in the case of the transport equation, this equation is also written in the conservative form. Consequently, solving successively for Eqs. (5) and (9) should preserve the shape of the hyperbolic tangent profile, and ensure the conservation of ψ, while transporting of the level set function ψ. Substituting Eq. (6) to Eq. (9) lead to another form of re-initialization equation [15,16]:
∂ψ/∂τ+∇.(ψ(1-ψ)n)=ϵ∇.(∇ψ) (10)
To develop a combined compact difference (CCD) scheme, the first and second derivatives must be solved simultaneously. Therefor using time splitting approach Eq. (10) is split as follows:
(ψ^*-ψ^n)/Δτ+∇.(ψ(1-ψ)n)=0 (11)
(ψ^(n+1)-ψ^*)/Δτ=ϵ∇.(∇ψ) (12)
A high-order compact finite difference approximation for the first derivative on a staggered mesh is given by [21,22]: α ̂f_(i-1)^'+f_i^'+α ̂f_(i+1)^'=a (f_(i+1/2)-f_(i-1/2))/h+b (f_(i+3/2)-f_(i-3/2))/3h (13) where ∂ψ/∂τ=f^'=∇f and to derive a fourth-order compact difference scheme, matching Taylor series coefficients lead to a=(9-6α ̂)/8 and b=(22α ̂-1)/8 for first derivatives.
A high-order compact finite difference approximation for the second derivative is given by [21,22]: α ̂g_(i-1)^"+g_i^"+α ̂g_(i+1)^"=a^' (g_(i+1)-2g_i+g_(i-1))/2h+b^' (g_(i+2)-2g_i+g_(i-2))/4h (14) where ∂ψ/∂τ=g^"=∇^2...

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...nal Physics. 155 (1999) 410–438.

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