Extrapolation in the Normal Direction

Abstract

In the last chapter we constructed signed distance functions by following characteristics that flow outward from the interface. Similar techniques can be used to propagate information in the direction of these characteristics. For example,

$$ {S_t} + \overrightarrow N \cdot \nabla S = 0 $$

(8.1) is a Hamilton-Jacobi equation (in S) that extrapolates S normal to the interface, i.e. so that S is constant on rays normal to the interface. Since $H\left( {\nabla S} \right) = \overrightarrow N \cdot \nabla S$, we can solve this equation with the techniques presented in Chapter 5 using H ₁ = n₁, H ₂ = n₂, and H ₃ = n ₃ .