Abstract
The main result of this thesis is to find a characteristic for conservation laws (CLaws) of explicit difference equations and prove that there is a one-to-one correspondence between the equivalence class of characteristics and conservation laws. This result enables us to prove the converse of Noether’s theorem for explicit difference equations: that there is a one-to-one correspondence between variational symmetries and conservation laws. Using the characteristic we show that the infinite hierarchy of conservation laws generated by the Gardner method for dpKdV are distinct. We use the characteristic to form a new method for finding CLaws of quad-graph equations that is similar to the existing method for searching for symmetries though more complicated. We apply this method to three quad-graph equations to find new five-point CLaws. The second focus of this thesis has been to develop a method for discretizing a scalar PDE, with polynomial nonlinearities, so as to locally preserve as many conservation laws as possible. This method is applied to the KdV equation resulting in new and known schemes that preserve the first and second CLaws together and the first and third CLaws together. These methods are then studied numerically and compared with a multisymplectic and a volume preserving scheme.