Reducing and monitoring round-off error propagation for symplectic implicit Runge-Kutta schemes
For researchers in numerical integration of Hamiltonian systems, this work provides an implementation with improved round-off error properties, though it is incremental.
The paper proposes an implementation of symplectic implicit Runge-Kutta schemes using fixed point iteration that is near-optimal in round-off error propagation for non-stiff Hamiltonian systems. It also presents a procedure to estimate round-off error via a second, less precise integration.
We propose an implementation of symplectic implicit Runge-Kutta schemes for highly accurate numerical integration of non-stiff Hamiltonian systems based on fixed point iteration. Provided that the computations are done in a given floating point arithmetic, the precision of the results is limited by round-off error propagation. We claim that our implementation with fixed point iteration is near-optimal with respect to round-off error propagation under the assumption that the function that evaluates the right-hand side of the differential equations is implemented with machine numbers (of the prescribed floating point arithmetic) as input and output. In addition, we present a simple procedure to estimate the round-off error propagation by means of a slightly less precise second numerical integration. Some numerical experiments are reported to illustrate the round-off error propagation properties of the proposed implementation.