Exponential Integrators Preserving First Integrals or Lyapunov Functions for Conservative or Dissipative Systems

Abstract

In this paper, combining the ideas of exponential integrators and discrete gradients, we propose and analyze a new structure-preserving exponential scheme for the conservative or dissipative systemẏ = Q(M y + ∇U (y)), where Q is a d × d skew-symmetric or negative semidefinite real matrix, M is a d × d symmetric real matrix, and U : R d → R is a differentiable function. We present two properties of the new scheme. The paper is accompanied by numerical results that demonstrate the remarkable superiority of our new scheme in comparison with other structure-preserving schemes in the scientific literature.

Exponential collocation methods for conservative or dissipative systems

“…[30,31,32,33,34]). In [37], the authors researched a new energy-preserving exponential scheme for the conservative or dissipative system. Here we note that its order is of only two since this scheme combines the ideas of DG and AVF methods.…”

“…Recently, in order to take advantage the structure of the underlying system and preserve its energy simultaneously, a novel energy-preserving method has been studied in [54,59] for secondorder ODEs and a new energy-preserving exponential scheme for the conservative or dissipative system has been researched in [37]. However, these two kinds of methods are both based on the AVF methods and thence they are only of order two, in general.…”

Exponential energy-preserving methods for charged-particle dynamics in a strong and constant magnetic field

“…Remark 2.2 It is noted that this kind of method belongs to exponential integrators, which have been widely developed and researched for solving highly oscillatory systems (see, e.g. [19,20,21,24,30,32]).…”

“…[10]) and trigonometric/exponential EP methods (see, e.g. [21,31,34]). Based on these work, we will derive and analyse novel EP integrators for charged-particle dynamics in a strong and constant magnetic field.…”

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

“…[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.…”