Computation of fluxes of conservation laws

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.