Reducing and monitoring round-off error propagation for symplectic implicit Runge-Kutta schemes
Copyright policy )Reducing and monitoring round-off error propagation for symplectic implicit Runge-Kutta schemes
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.
Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems, symplectic implicit Runge-Kutta methods, round-off errors, Numerical Analysis (math.NA), Numerical methods for Hamiltonian systems including symplectic integrators, Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations, stopping criterion, FOS: Mathematics, fixed-point iteration, Mathematics - Numerical Analysis, non-stiff Hamiltonian systems, numerical experiments
Discretization methods and integrators (symplectic, variational, geometric, etc.) for dynamical systems, symplectic implicit Runge-Kutta methods, round-off errors, Numerical Analysis (math.NA), Numerical methods for Hamiltonian systems including symplectic integrators, Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations, stopping criterion, FOS: Mathematics, fixed-point iteration, Mathematics - Numerical Analysis, non-stiff Hamiltonian systems, numerical experiments
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