Numerical methods for high dimensional Hamilton–Jacobi equations using radial basis functions

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.