Characteristics of Conservation Laws for Difference Equations

• Published in: Foundations of computational mathematics Vol. 13; no. 4; pp. 667 - 692
• Main Authors: Grant, Timothy J., Hydon, Peter E.
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.08.2013; Springer; Springer Nature B.V
• Subjects: Applications of Mathematics; Computational mathematics; Computer Science; Conservation laws; Conservation laws (Physics); Derivation; Derivatives; Difference equations; Differential equations, Partial; Economics; Linear and Multilinear Algebras; Lotka-Volterra equations; Math Applications in Computer Science; Mathematical analysis; Mathematical models; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Partial differential equations; Theorems; United Kingdom; Conservation laws; Noether’s Theorem; Difference equations; 37K05; 12H10; 39A14
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-013-9151-2

Each conservation law of a given partial differential equation is determined (up to equivalence) by a function known as the characteristic. This function is used to find conservation laws, to prove equivalence between conservation laws, and to prove the converse of Noether’s Theorem. Transferring these results to difference equations is nontrivial, largely because difference operators are not derivations and do not obey the chain rule for derivatives. We show how these problems may be resolved and illustrate various uses of the characteristic. In particular, we establish the converse of Noether’s Theorem for difference equations, we show (without taking a continuum limit) that the conservation laws in the infinite family generated by Rasin and Schiff are distinct, and we obtain all five-point conservation laws for the potential Lotka–Volterra equation.