Computation of fluxes of conservation laws

TL;DR

This paper reviews methods for computing fluxes of conservation laws in PDEs, comparing approaches, illustrating with examples, and discussing symbolic software implementations for the direct method of conservation law construction.

Contribution

It provides a comprehensive comparison of flux computation methods within the direct conservation law construction framework, including practical software implementation details.

Findings

Multiple flux computation methods are compared and analyzed.

Examples illustrate the application of different methods.

Implementation in symbolic software enhances practical usability.

Abstract

The direct method of construction of local conservation laws of partial differential equations (PDE) is a systematic method applicable to a wide class of PDE systems [Anco S. and Bluman G., Direct construction method for conservation laws of partial differential equations Part II: General treatment. {\sl Europ. J. Appl. Math.} {\bf 13}, 567--585 (2002)]. Within the direct method, one seeks multipliers, such that the linear combination of PDEs of a given system with these multipliers yields a divergence expression. After local conservation law multipliers are found, one needs to reconstruct the fluxes of the conservation law. In this review paper, we discuss common methods of flux computation, compare them, and illustrate by examples. An implementation of these methods in symbolic software is also presented.