Projected exponential Runge–Kutta methods for preserving dissipative properties of perturbed constrained Hamiltonian systems

Abstract

Preserving conservative and dissipative properties of dynamical systems is desirable in numerical integration. To this end, we develop and implement numerical methods that preserve the exact rate of dissipation in certain qualitative properties of dissipatively perturbed constrained Hamiltonian systems, which are shown to be conformal symplectic. Projection methods based on exponential Runge–Kutta methods are proposed for such systems. These numerical schemes are shown to be constraint preserving, conformal invariants preserving, symmetric, and second-order accurate. It is shown that these structure-preserving methods can be further composed to obtain higher-order structure-preserving methods. Linear stability analysis is used to derive stability properties and conformal preservation of the phase-space area. Numerical experiments, including constrained oscillators and a damped Korteweg–de Vries partial differential equation, demonstrate the advantages of geometric integration and verify theoretical results.