Journal of Scientific Computing

, Volume 71, Issue 1, pp 144–165 | Cite as

Solving 1D Conservation Laws Using Pontryagin’s Minimum Principle

  • Wei Kang
  • Lucas C. Wilcox


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 space-independent 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 Lax–Oleinik 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 space-dependent and space-independent 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 space-independent flux functions.


Conservation laws Pontryagin’s minimum principle Spectral method Burgers’ equation 



This work was supported in part by AFOSR, NRL, and DARPA.


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Copyright information

© Springer Science+Business Media New York (outside the USA) 2016

Authors and Affiliations

  1. 1.Department of Applied MathematicsNaval Postgraduate SchoolMontereyUSA

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