Mathematics Colloquia and Seminars

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Let z(s) be a two times continuously differentiable curve on a bounded, simply connected domain Omega in the plane that separates two materials or fluids, say materials 1 and 2. Cover Omega with a uniform square computational grid of side h and suppose that the only information one is given about the curve z is the area fraction 0 <= Lambda_ij <= 1 that is occupied by material 1 in each cell C_ij of the grid. I will show how to construct an approximation to the curve z using a single line segment in each cell which is second-order accurate in the max norm and outline a proof of this fact.

Numerical methods for approximating a curve or a surface in three dimensions on a computational grid that are based solely on the volume fraction information associated with the curve are known as volume-of-fluid (VOF) methods. The problem I have outlined above is known as the "interface reconstruction problem" for a VOF method. Besides being the first proof that an algorithm for solving this problem converges to the given interface z, this result is interesting because it provides a criterion for determining whether the reconstructed interface is "well-resolved" on a given grid. This criterion -- which depends only on the curvature kappa(s) of the initial data z(s) in the 3 x 3 block cells centered on the cell C_ij -- is surprising, since it is of the form sqrt(h) <= 1 / kappa rather than h <= C / kappa where C is some positive constant that is independent of h. In other words, in order to accurately approximate curves that have "hairpin" turns at very small scales it is necessary to have a much finer grid; namely one that decreases like the square of the curvature rather than linearly with the curvature, as one might initially expect. In closing I will indicate how this constraint on the size of the grid may apply to ANY algorithm for accurately approximating a given curve z on a computational grid, including level set methods, the immersed boundary method, the immersed interface method, etc.

Gerry's Birthday