## The multiplier method to construct conservative finite difference schemes for ordinary and partial differential equations

### Andy Wan

McGill University

Structure-preserving discretizations are numerical methods which preserve important structure of differential equations at the discrete level. For system derived from a Hamiltonian or Lagrangian, symplectic and variational integrators are a class of discretizations which can preserve symplectic and variational structure at the discrete level. In this talk, we introduce the multiplier method of constructing conservative finite difference schemes for ordinary and partial differential equations. The proposed discretization is shown to be consistent for any order of accuracy when the discrete multiplier has a multiplicative inverse. Moreover, we show that by construction, the discrete densities can be exactly conserved. In particular, the multiplier method does not require the system to possess a symplectic or variational structure. Examples, including dissipative problems, are given to illustrate the method. In the case when the inverse of the discrete multiplier becomes singular, consistency is also established for scalar ODEs provided the discrete multiplier and density are zero-compatible. Long-term stability for such method is also discussed. This is joint work with Alexander Bihlo at Memorial University and Jean-Christophe Nave at McGill University.