Speaker
Description
This talk considers Poisson systems with a Hamiltonian describing the energy in the system. If the Hamiltonian is quadratic, the energy is exactly conserved using the Gauss-Legendre methods as numerical integrators. Since Gauss methods are implicit Runge-Kutta methods, a system of equations must be solved at each time step. The computation can be very expensive for high-dimensional systems which asks for fast structure-preserving iterative solvers. In the linear case, the Gauss methods can be reformulated via their respective stability functions. These stability functions coincide with the diagonal Padé approximations of the exponential function. They can be understood as matrix-valued functions that map elements of a Lie algebra to a quadratic Lie group, which characterizes the conservation of energy. When it comes to evaluation, there are two options. Either the Padé approximation is computed directly or a further reformulation using the idea of partial fraction decomposition is considered. In both cases, one or more systems of linear equations have to be solved because inverse matrices appear. If the underlying matrix of the Possion system is sparse, Krylov subspace methods are a good choice. A special variation of a Krylov subspace method is an Arnoldi approximation. We show that the Arnoldi method through a clever choice of the basis of the Krylov subspace, leads to energy-conserving iterations.
