Exponential collocation methods for conservative or dissipative systems

In this paper, we propose and analyse a novel class of exponential collocation methods for solving conservative or dissipative systems based on exponential integrators and collocation methods. It is shown that these novel methods can be of arbitrarily high order and exactly or nearly preserve first integrals or Lyapunov functions. We also consider order estimates of the new methods. Furthermore, we explore and discuss the application of our methods in important stiff gradient systems, and it turns out that our methods are unconditionally energy-diminishing and strongly damped even for very stiff gradient systems. Practical examples of the new methods are derived and the efficiency and superiority are confirmed and demonstrated by three numerical experiments including a nonlinear Schr\"{o}dinger equation. As a byproduct of this paper, arbitrary-order trigonometric/RKN collocation methods are also presented and analysed for second-order highly oscillatory/general systems. The paper is accompanied by numerical results that demonstrate the great potential of this work.