A second order accurate level set method on non-graded adaptive cartesian grids

  • Authors:

  • Chohong Min;Frédéric Gibou

  • Affiliations:

  • Department of Mathematics, KyungHee University, Korea;Mechanical Engineering Department, University of California, Santa Barbara, CA 93106, United States and Computer Science Department, University of California, Santa Barbara, CA 93106, United State ...

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2007

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Abstract

We present a level set method on non-graded adaptive Cartesian grids, i.e. grids for which the ratio between adjacent cells is not constrained. We use quadtree and octree data structures to represent the grid and a simple algorithm to generate a mesh with the finest resolution at the interface. In particular, we present (1) a locally third order accurate reinitialization scheme that transforms an arbitrary level set function into a signed distance function, (2) a second order accurate semi-Lagrangian methods to evolve the linear level set advection equation under an externally generated velocity field, (3) a second order accurate upwind method to evolve the non-linear level set equation under a normal velocity as well as to extrapolate scalar quantities across an interface in the normal direction, and (4) a semi-implicit scheme to evolve the interface under mean curvature. Combined, we obtain a level set method on adaptive Cartesian grids with a negligible amount of mass loss. We propose numerical examples in two and three spatial dimensions to demonstrate the accuracy of the method.