A Level Set Method for the Computation of Multivalued Solutions to Quasi-linear Hyperbolic Pdes and Hamilton-jacobi Equations

We develop a level set method for the computation of multivalued solutions to quasi-linear hyperbolic partial differential equations and Hamilton-Jacobi equations in any number of space dimensions. We use the classic idea of Courant and Hilbert to define the solution of the quasi-linear hyperbolic PDEs or the gradient of the solution to the Hamilton-Jacobi equations as zero level sets of level set functions. Then the evolution equations for the level set functions satisfy linear Liouville equations defined in the ”phase” space, unfolding the singularities and preventing the numerical capturing of the viscosity solution. This provides a computational framework for the computations of multivalued geometric solutions to general quasilinear PDEs. By using the local level set method the cost of each time update for this method is O(N logN) for a d dimensional problem, where N is the number of grid points in each dimension.