Speaker: Brynjulf Owren
Norwegian University of Science and Technology Trondheim, Norway
Title: Integral Preserving Numerical Integrators for PDEs
Abstract: We consider partial differential equations that can be written in the form u_t = D dH/du
(1) where H[u] is a first integral, for example a Hamiltonian functional, and D is a skew-adjoint operator which may depend on u, but not necessarily define a Poisson bracket. A fundamental type of numerical time integrator for (1) is one that preserves H exactly for all times, and its construction is well-known. This type of scheme is usually implicit, and a consequence of this is that one needs to solve an algebraic system exactly (or to machine precision) in every time step in order to maintain the preservation of H. For this reason, we shall here propose a procedure which will, in the case that H is an integral of a polynomial function in u, result in a scheme that is only linearly implicit. This means that the solution of one linear system in each time step suffices to obtain a conservative scheme. The procedure is based on a polarization technique, in which H is replaced by a multivariate function H such that H[u] = H[u,...,u]. We investigate how the invariance of H under the cyclic group of permutations not only ensures that the resulting integrator is conservative, but also that its formal order of consistency is at least two.
Finally we shall comment on the spatial discretization of the PDE and consider the discrete Euler-Lagrange operator for the new procedure and its properties.