Peter E. Hydon

Gianluca Frasca-Caccia, Peter E. Hydon

Conservation laws are among the most fundamental geometric properties of a partial differential equation (PDE), but few known finite difference methods preserve more than one conservation law. All conservation laws belong to the kernel of the Euler operator, an observation that was first used recently to construct approximations symbolically that preserve two conservation laws of a given PDE. However, the complexity of the symbolic computations has limited the effectiveness of this approach. The current paper introduces some key simplifications that make the symbolic-numeric approach feasible. To illustrate the simplified approach, we derive bespoke finite difference schemes that preserve two discrete conservation laws for the Korteweg-de Vries (KdV) equation and for a nonlinear heat equation. Numerical tests show that these schemes are robust and highly accurate compared to others in the literature.

Timothy J. Grant, Peter E. Hydon

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 that the infinite family of conservation laws generated by Rasin and Schiff using the Gardner transformation are distinct (without taking a continuum limit), and we obtain all five-point conservation laws for the potential Lotka-Volterra equation.

Linyu Peng, Peter E. Hydon

The difference variational bicomplex, which is the natural setting for systems of difference equations, is constructed and used to examine the geometric and algebraic properties of various systems. Exactness of the bicomplex gives a coordinate-free setting for finite difference variational problems, Euler--Lagrange equations and Noether's theorem. We also examine the connection between the condition for the existence of a Hamiltonian and the multisymplecticity of systems of partial difference equations. Furthermore, we define difference multimomentum maps of multisymplectic systems, which yield their conservation laws. To conclude, we adapt the variational bicomplex to multisymplectic integrators on a mesh that is logically rectangular. By scaling horizontal forms and difference operators according to the local step sizes, all of the results derived earlier can be applied, whether or not the mesh is uniform.