Solving 1D Conservation Laws Using Pontryagin's Minimum Principle

TL;DR

This paper introduces a novel control-based method using Pontryagin's minimum principle to solve scalar convex conservation laws, extending classical formulas and providing an algorithm with numerical demonstrations.

Contribution

It establishes a connection between conservation laws and optimal control, generalizes the Lax--Oleinik formula, and offers a practical algorithm with MATLAB implementation for solving these laws.

Findings

The proposed algorithm accurately approximates solutions to scalar convex conservation laws.

Numerical examples demonstrate the method's effectiveness for space-dependent and space-independent cases.

A MATLAB routine with Chebfun is provided for practical implementation.

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 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.