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Solving 1D Conservation Laws Using Pontryagin’s Minimum Principle

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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.

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Notes

  1. Chebfun is a MATLAB [27] package that allows symbolic-like manipulation of functions at numerics speed using Chebyshev and Fourier series [3].

  2. Chebfun with a capital C is the name of the software package, chebfun with a lowercase c is a single variable function defined on an interval created with the package [3].

  3. More specifically, we used Clawpack v5.3.1-11-geb31727 from https://github.com/clawpack/clawpack with the Chapter 11 examples in the git repository https://github.com/clawpack/apps (git commit ba557b49852377c05192d48289b3fbc8fea0f52e).

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Acknowledgments

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

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  1. Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, USA

    Wei Kang & Lucas C. Wilcox

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  1. Wei Kang
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  2. Lucas C. Wilcox
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Correspondence to Lucas C. Wilcox.

Appendix: Example Using Implementation 1

Appendix: Example Using Implementation 1

As an example of using Implementation 1, we present the code used in Example 1 to generate Fig. 1.

figure f

Here we give the point-wise solver to to generate polynomial approximations of the solution, , for various time instances, .

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Kang, W., Wilcox, L.C. Solving 1D Conservation Laws Using Pontryagin’s Minimum Principle. J Sci Comput 71, 144–165 (2017). https://doi.org/10.1007/s10915-016-0294-6

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