Solving 1D Conservation Laws Using Pontryagin's Minimum Principle
Abstract
This paper discusses a connection between scalar convex conservation laws and Pontryagin's minimum principle. For flux functions for which an associated optimal control problem can be found, a minimum value solution of the conservation law is proposed. For scalar spaceindependent convex conservation laws such a control problem exists and the minimum value solution of the conservation law is equivalent to the entropy solution. This can be seen as a generalization of the LaxOleinik formula to convex (not necessarily uniformly convex) flux functions. Using Pontryagin's minimum principle, an algorithm for finding the minimum value solution pointwise of scalar convex conservation laws is given. Numerical examples of approximating the solution of both spacedependent and spaceindependent conservation laws are provided to demonstrate the accuracy and applicability of the proposed algorithm. Furthermore, a MATLAB routine using Chebfun is provided (along with demonstration code on how to use it) to approximately solve scalar convex conservation laws with spaceindependent flux functions.
 Publication:

arXiv eprints
 Pub Date:
 May 2016
 arXiv:
 arXiv:1605.04473
 Bibcode:
 2016arXiv160504473K
 Keywords:

 Mathematics  Numerical Analysis;
 Mathematics  Optimization and Control;
 35L65;
 65M70;
 65M25;
 49L25;
 49M05