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Coupling Numerical Schemes for Hyperbolic Equations to Interface Evolution
Central Idea:
Once the
link between evolving fronts and hyperbolic conservation
laws was established,
Ref. 1 suggests the use of the numerical technology of shock schemes
to move the interface.
First, the equation of motion
for a propagating curve which remains a graph as it moves
is written as a HamiltonJacobi equation. It is formally
shown that this is a hyperbolic conservation law with viscous
righthandside. As discussed in Ref. 1 (Sethian, 1987):


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In the special case of a curve moving normal to itself with speed F = 1  epsilon k, where epsilon is small number and "k" is the curvature, the equations of motion become a HamiltonJacobi equation with viscous righthandside Such equations are part of a general theory of viscosity solutions developed by Crandall and Lions in Refs. 2 and 3 below. This theory develops, for HamiltonJacobi equations of the form u_t + H(x,y,z,u_x,u_y,u_z) = 0, a rigorous analysis of what happens once singularities develop in the solution. This theory mirrors that for hyperbolic conservation laws, and provides theoretical underpinnings for some of the work on level set methods and Fast Marching Methods .
The next step is then to exploit all the previous work and now consider interfaces which do not remain graphs as they evolve.. At the same time, considerable theoretical work on viscosity solutions of HamiltonJacobi equations comes into play.
AbstractIn many physical problems, a key aspect is the motion of a propagating front separating two components. As fundamental as this may be, the development of a numerical algorithm to accurately track the moving front is difficult. In this report, we describe some previous theoretical and numerical work. We begin with two examples to motivate the problem, followed by some analytical results. These theoretical results are then used as a foundation for two different types of numerical schemes. Finally, we describe the application of one of these schemes to our work in combustion.