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
