Conservation laws are ubiquitous in the study of differential equations. They often represent fundamental physical quantities, and can lead to stability and existence results for possible solutions. We researched the concept of conservation laws for discrete difference equations, such as those that arise from finite difference discretizations of partial differential equations. The goal was to study the relationship between such discrete conservation laws and their continuous counterparts. We approached this by considering the variational bicomplex, a double chain complex which in the smooth case is defined on the infinite jet bundle of a fibered manifold, and provides a natural algebraic setting for studying conservation laws. By constructing an appropriate counterpart structure over a discrete space, we aimed to be able to fundamentally understand discrete conservation laws and moreover contrast the smooth and discrete theory.

Funded by the McGill SURA grant during the summer of 2024. Continued for course credit as part of McGill’s MATH470 (Honours Research Project) course.