A Fast and Accurate Semi-Lagrangian Particle Level Set Method

In this paper, we present an efficient semi-Lagrangian based particle level set method for the accurate capturing of interfaces. This method retains the robust topological properties of the level set method without the adverse effects of numerical dissipation. Both the level set method and the particle level set method typically use high order accurate numerical discretizations in time and space, e.g. TVD RungeKutta and HJ-WENO schemes. We demonstrate that these computationally expensive schemes are not required. Instead, fast, low order accurate numerical schemes suffice. That is, the addition of particles to the level set method not only removes the difficulties associated with numerical diffusion, but also alleviates the need for computationally expensive high order accurate schemes. We use an efficient, first order accurate semi-Lagrangian advection scheme coupled with a first order accurate fast marching method to evolve the level set function. To accurately track the underlying flow characteristics, the particles are evolved with a second order accurate method. Since we avoid complex high order accurate numerical methods, extending the algorithm to arbitrary data structures becomes more feasible, and we show preliminary results obtained with an octree-based adaptive mesh. ∗Research supported in part by an ONR YIP and PECASE award (N00014-01-1-0620), a Packard Foundation Fellowship, a Sloan Research Fellowship, ONR N00014-03-1-0071, ONR N00014-02-1-0720, NSF DMS-0106694, NSF ITR-0121288 and DOE under the ASCI Academic Strategic Alliances Program (LLNL contract B341491). In addition, the first author was supported in part by an NSF postdoctoral fellowship (DMS-0202459). †Mathematics Department, UCLA, Los Angeles, CA 90095. ‡Computer Science Department, Stanford University, Stanford, CA 94305.