We utilize radial basis functions (RBFs) to construct numerical schemes for Hamilton-Jacobi (HJ) equations on unstructured data sets in arbitrary dimensions. The computational setup is a meshless discretization of the physical domain. We derive monotone schemes on unstructured data sets to compute the viscosity solutions. The essentially nonoscillatory (ENO) mechanism is combined with radial basis function reconstruction to obtain high order schemes in the presence of gradient discontinuities. Numerical examples of time dependent HJ equations in 2, 3 and 4 dimensions illustrate the accuracy of the new methods. (C) 2003 Elsevier Inc. All rights reserved.

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## Scholarly Works (13 results)

We propose a simple, fast sweeping method based on the Lax-Friedrichs monotone numerical Hamiltonian to approximate viscosity solutions of arbitrary static Hamilton-Jacobi equations in any number of spatial dimensions. By using the Lax-Friedrichs numerical Hamiltonian, we can easily obtain the solution at a specific grid point in terms of its neighbors, so that a Gauss-Seidel type nonlinear iterative method can be utilized. Furthermore, by incorporating a group-wise causality principle into the Gauss-Seidel iteration by following a finite group of characteristics, we have an easy-to-implement, sweeping-type, and fast convergent numerical method. However, unlike other methods based on the Godunov numerical Hamiltonian, some computational boundary conditions are needed in the implementation. We give a simple recipe which enforces a version of discrete min-max principle. Some convergence analysis is done for the one-dimensional eikonal equation. Extensive 2-D and 3-D numerical examples illustrate the efficiency and accuracy of the new approach. To our knowledge, this is the first fast numerical method based on discretizing the Hamilton-Jacobi equation directly without assuming convexity and/or homogeneity of the Hamiltonian. (C) 2003 Elsevier Inc. All rights reserved.