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, satisfylinear 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:**- Available as PostScript (7.5 Mbytes) or gzipped PostScript (440 Kbytes; uncompress using gunzip).
**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.

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