Numerical preservation of multiple local conservation laws

There are several well-established approaches to constructing finite difference schemes that preserve global invariants of a given partial differential equation. However, few of these methods preserve more than one conservation law locally. A recently-introduced strategy uses symbolic algebra to construct finite difference schemes that preserve several local conservation laws of a given scalar PDE in Kovalevskaya form. In this paper, we adapt the new strategy to PDEs that are not in Kovalevskaya form and to systems of PDEs. The Benjamin–Bona–Mahony equation and a system equivalent to the nonlinear Schrödinger equation are used as benchmarks, showing that the strategy yields conservative schemes which are robust and highly accurate compared to others in the literature.