Preprint 2003-025

A Level Set Method for the Computation of Multivalued Solutions to Quasi-Linear Hyperbolic PDEs and Hamilton-Jacobi Equations

Abstract: 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. The idea is 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. We then derive the evolution equations for the level set functions which, surprisingly, satisfy linear Liouville equations. By using the local level set method the cost of each time update for this method is $O(N^d \log N)$ for a $d$ dimensional problem, where $N$ is the number of grid points in each dimension.

Paper:

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Author(s):

Shi Jin, <jin@math.wisc.edu>

Stanley J. Osher, <sjo@math.ucla.edu>

Publishing information:

Communications in Mathematical Sciences, submitted

Comments:

Submitted by:

<jin@math.wisc.edu> April 7 2003.