## 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.

Original language | English |
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Pages (from-to) | 300-321 |

Number of pages | 22 |

Journal | Journal of Computational Physics |

Volume | 225 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jul 2007 |

### Bibliographical note

Funding Information:The research of F. Gibou was supported in part by the Alfred P. Sloan Foundation through a research fellowship in Mathematics.

## Keywords

- Adaptive mesh refinement
- Extrapolation in the normal direction
- Ghost fluid method
- Level set method
- Motion by mean curvature
- Motion in an externally generated velocity field
- Motion in the normal direction
- Non-graded Cartesian grids