Constraint and Structure Preservation in Pde

In this set of notes we examine numerical techniques for preservation of constraints and (geometric) structures in ODE and PDE systems, with application to the Einstein equations. The techniques are based on explicit enforcement of constraints using Lagrange multiplier methods, and hence involve a type of (controlled) projection onto the constraint manifold. The resulting numerical methods always have the following two properties: 1) they produce solutions which are comparable in accuracy to standard methods which do not enforce the constraints, and 2) they enforce the constraints (exactly). They can sometimes be shown to have an additional property, namely 3) they preserve geometric structure such as time-reversibility and symplecticity. The numerical techniques for the ODE case can be found in the literature on constrained molecular dynamics as far back as the early 1990’s, but the PDE case has not been completely developed. We use Lagrangian and Hamiltonian formalism for mechanical (finite-dimensional) and field (infinite-dimensional) systems throughout these notes, but also apply the techniques to more general non-variational problems with constraints. In the last section we consider application to various constrained formulations of the Einstein equations.