Quantum mechanics

Superconvergence

Social psychology

Differential equation

Applied mathematics

Hamiltonian system

Differential algebraic equation

Mathematical optimization

Hamiltonian (control theory)

Numerical methods for ordinary differential equations

Thermodynamics

Physics

Finite element method

Psychology

Mathematics

Relaxation (psychology)

Conservation law

Mathematical analysis

Conservation of energy

Ordinary differential equation

Runge–Kutta methods

Journal of Scientific Computing

The recently-introduced relaxation approach for Runge–Kutta methods can be used to enforce conservation of energy in the integration of Hamiltonian systems. We study the behavior of implicit and explicit relaxation Runge–Kutta methods in this context. We find that, in addition to their useful conservation property, the relaxation methods yield other improvements. Experiments show that their solutions bear stronger qualitative similarity to the true solution and that the error grows more slowly in time. We also prove that these methods are superconvergent for a certain class of Hamiltonian systems.

Relaxation Runge–Kutta Methods for Hamiltonian Problems