Nonconvex Hamilton Jacobi equations - Adam Oberman Math

This is a project by

Adam Oberman Main Page

based on work by

Lawrence (Craig) Evans, Mathematics, University of California at Berkeley, http://math.berkeley.edu/~evans/

Introduction

we are going to attempt to better understand the behavior of solutions of the (forward in time) initial value problem for the nonconvex Hamilton Jacobi equation

$u t + H (D u) = 0$

with initial data

$u (x,0) = g (x)$

by studying (solving) the (backward in time) final value problem for the measure

$σ t - d i v (σ D H (D u) = 0$

with final values

$σ(x, T) = γ(x)$

Plan for the numerics

First, understand and recover the weak version of the method of characteristics, by solving the adjoint equation in the linear, constant coefficient case. This is just to make sure that we can track the characteristic, $x'(t) = c$ , where the equation is $u t = c u x$

Next, same thing with non-constant characteristics $u t = c (x) u x$

Next, better to do conservation laws, or straight to HJ equations? Conservation laws are in the form

$v t = H'(v) v x$ , which is quasilinear. Doesn't matter too much, since the adjoint equation is still linear, but tracking characteristics is more clear in this case.

Then do convex HJ, $u t = c | u x |$ and concave, seperately, to see the different kinds of shocks.

Finally, non-convex (non-concave) example.

Numerical Method for the Advection

Good to learn in general:

Referece: Osher Fedkiw chapter 5, HJ Equations, page 50, Lax-Freidrichs.