Exponential collocation methods for conservative or dissipative systems

Abstract

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 energydiminishing 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 Schrödinger 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.

Arbitrary-order functionally fitted energy-diminishing methods for gradient systems

“…In order to formulate the novel methods, we will use the functionally fitted technology, which is a popular approach to constructing efficient and effective methods in scientific computing (see, e.g. [11,22]). To this end, define a function space Y =span{ϕ 0 (t), .…”

“…[2,5]). Many effective methods have been derived for this stiff gradient system with a constant matrix G and we refer to [7,8,10,12,21,22,23,24,25] for example. The FFED method (2) for solving this stiff gradient system is defined as follows.…”

“…This scheme has been presented and researched in [22]. Rewrite this scheme as y 1 = R(−hA)y 0 with the stability function R(−hA) = e −hA .…”

Symmetry Groups, Similarity Reductions, and Conservation Laws of the Time-Fractional Fujimoto–Watanabe Equation Using Lie Symmetry Analysis Method

“…e exploration of solutions for NFDEs is a crucial aspect. A variety of methods are presented, for instance, the first integral method [25], functional variable method [26], auxiliary equation method [27], and exponential function method [28].…”

Exponential collocation methods based on continuous finite element approximations for efficiently solving the cubic Schrödinger equation

“…This class of methods has also been studied in the numerical integration of Schrödinger equations (see, e.g., [11, 23–26]). Recently, a novel kind of exponential integrators is developed and analyzed for ODEs in [67]. However, it seems that until now, exponential integrators with a favorable continuous energy preservation for Schrödinger equations have not been studied in the literature.…”