In this paper, we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field. Our method uses Lagrangian marker particles to rebuild the level set in regions which are under-resolved. This is often the case for flows undergoing stretching and tearing. The overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method. Our method compares favorably with volume of fluid methods in the conservation of mass and purely Lagrangian schemes for interface resolution. The method is presented in three spatial dimensions.

Abstract-We describe and implement an algorithm for computing the set of reachable states of a continuous dynamic game. The algorithm is based on a proof that the reachable set is the zero sublevel set of the viscosity solution of a particular time-dependent Hamilton-Jacobi-Isaacs partial differential equation. While alternative techniques for computing the reachable set have been proposed, the differential game formulation allows treatment of nonlinear systems with inputs and uncertain parameters. Because the time-dependent equation's solution is continuous and defined throughout the state space, methods from the level set literature can be used to generate more accurate approximations than are possible for formulations with potentially discontinuous solutions. A numerical implementation of our formulation is described and has been released on the web. Its correctness is verified through a two vehicle, three dimensional collision avoidance example for which an analytic solution is available.

We describe a set of best practices for scientific software development, based on research and experience, that will improve scientists' productivity and the reliability of their software.

Hybrid system theory lies at the intersection of the fields of engi-

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Abstract. Using only the existence and uniqueness of trajectories for a generic dynamic system with inputs, we dene and examine eight types of forward and backward reachability constructs. If the input is treated in a worst-case fashion, any forward or backward reach set or tube can be used for safety analysis, but if the input is treated in a best-case fashion only the backward reach tube always provides the correct results. Fortunately, forward and backward algorithms can be exchanged if wellposed reverse time trajectories can be dened. Unfortunately, backward reachability constructs are more likely to suer from numerical stability issues, especially in systems with signicant contraction|the very systems where forward simulation and reachability are most eective.