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

A long-term numerical energy-preserving analysis of symmetric and/or symplectic extended RKN integrators for efficiently solving highly oscillatory Hamiltonian systems

The primary objective of this paper is to present a long-term numerical energy-preserving analysis of one-stage explicit symmetric and/or symplectic extended Runge--Kutta--Nystr\"{o}m (ERKN) integrators for highly oscillatory Hamiltonian systems. We study the long-time numerical energy conservation not only for symmetric integrators but also for symplectic integrators. In the analysis, we neither assume symplecticity for symmetric methods, nor assume symmetry for symplectic methods. It turns out that these both kinds of ERKN integrators have a near conservation of the total and oscillatory energy over a long term. To prove the result for symmetric integrators, a relationship between symmetric ERKN integrators and trigonometric integrators is established by using Strang splitting and based on this connection, the long-time conservation is derived. For the long-term analysis of symplectic ERKN integrators, the above approach does not work anymore and we use the technology of modulated Fourier expansion developed in SIAM J. Numer. Anal. 38 (2000) by Hairer and Lubich. By taking some novel adaptations of this essential technology for non-symmetric methods, we derive the modulated Fourier expansion for symplectic ERKN integrators. Moreover, it is shown that the symplectic ERKN integrators have two almost-invariants and then the near energy conservation over a long term is obtained

Long-time oscillatory energy conservation of total energy-preserving methods for highly oscillatory Hamiltonian systems

For an integrator when applied to a highly oscillatory system, the near conservation of the oscillatory energy over long times is an important aspect. In this paper, we study the long-time near conservation of oscillatory energy for the adopted average vector field (AAVF) method when applied to highly oscillatory Hamiltonian systems. This AAVF method is an extension of the average vector field method and preserves the total energy of highly oscillatory Hamiltonian systems exactly. This paper is devoted to analysinganother important property of AAVF method, i.e., the near conservation of its oscillatory energy in a long term. The long-time oscillatory energy conservation is obtained via constructing a modulated Fourier expansion of the AAVF method and deriving an almost invariant of the expansion. A similar result of the method in the multi-frequency case is also presented in this paper

Long-term analysis of symplectic or symmetric extended RKN methods for nonlinear wave equations

This paper analyses the long-time behaviour of one-stage symplectic or symmetric extended Runge--Kutta--Nystr\"{o}m (ERKN) methods when applied to nonlinear wave equations. It is shown that energy, momentum, and all harmonic actions are approximately preserved over a long time for one-stage explicit symplectic or symmetric ERKN methods when applied to nonlinear wave equations via spectral semi-discretisations. For the long-term analysis of symplectic or symmetric ERKN methods, we derive a multi-frequency modulated Fourier expansion of the ERKN method and show three almost-invariants of the modulation system. In the analysis of this paper, we neither assume symmetry for symplectic methods, nor assume symplecticity for symmetric methods. The results for symplectic and symmetric methods are obtained as a byproduct of the above analysis. We also give another proof by establishing a relationship between symplectic and symmetric ERKN methods and trigonometric integrators which have been researched for wave equations in the literature

Dynamics, numerical analysis, and some geometry

Geometric aspects play an important role in the construction and analysis of structure-preserving numerical methods for a wide variety of ordinary and partial differential equations. Here we review the development and theory of symplectic integrators for Hamiltonian ordinary and partial differential equations, of dynamical low-rank approximation of time-dependent large matrices and tensors, and its use in numerical integrators for Hamiltonian tensor network approximations in quantum dynamics.Comment: prepared for the Proceedings of ICM 2018 (Christian Lubich's plenary talk

Functionally-fitted energy-preserving integrators for Poisson systems

In this paper, a new class of energy-preserving integrators is proposed and analysed for Poisson systems by using functionally-fitted technology. The integrators exactly preserve energy and have arbitrarily high order. It is shown that the proposed approach allows us to obtain the energy-preserving methods derived in BIT 51 (2011) by Cohen and Hairer and in J. Comput. Appl. Math. 236 (2012) by Brugnano et al. for Poisson systems. Furthermore, we study the sufficient conditions that ensure the existence of a unique solution and discuss the order of the new energy-preserving integrators.Comment: 19 page

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

In this paper, exponential energy-preserving methods are formulated and analysed for solving charged-particle dynamics in a strong and constant magnetic field. The resulting method can exactly preserve the energy of the dynamics. Moreover, it is shown that the magnetic moment of the considered system is nearly conserved over a long time along this exponential energy-preserving method, which is proved by using modulated Fourier expansions. Other properties of the method including symmetry and convergence are also studied. An illustrated numerical experiment is carried out to demonstrate the long-time behaviour of the method

Error analysis of one-stage explicit extended RKN integrators for semilinear wave equations

In this paper, we present an error analysis of one-stage explicit extended Runge--Kutta--Nystr\"{o}m integrators for semilinear wave equations. These equations are analysed by using spatial semidiscretizations with periodic boundary conditions in one space dimension. Optimal second-order convergence is proved without requiring Lipschitz continuous and higher regularity of the exact solution. Moreover, the error analysis is not restricted to the spectral semidiscretization in space

Long-time energy analysis of extended RKN integrators for muti-frequency highly oscillatory Hamiltonian systems

In this paper, we study the long-time near conservation of the total and oscillatory energies for extended RKN (ERKN) integrators when applied to muti-frequency highly oscillatory Hamiltonian systems. We consider one-stage explicit symmetric integrators and show their long-time behaviour of numerical energy conservations by using modulated multi-frequency Fourier expansions. Numerical experiments are carried out and the numerical results demonstrate the remarkable long-time near conservation of the energies for the ERKN integrators and support our theoretical analysis presented in this paper

Diagonal implicit symplectic ERKN methods for solving oscillatory Hamiltonian systems

This paper studies diagonal implicit symplectic extended Runge--Kutta--Nystr\"{o}m (ERKN) methods for solving the oscillatory Hamiltonian system $H(q,p)=\dfrac{1}{2}p^{T}p+\dfrac{1}{2}q^{T}Mq+U(q)$. Based on symplectic conditions and order conditions, we construct some diagonal implicit symplectic ERKN methods. The stability of the obtained methods is discussed. Three numerical experiments are carried out and the numerical results demonstrate the remarkable numerical behavior of the new diagonal implicit symplectic methods when applied to the oscillatory Hamiltonian system.Comment: 17 pages, 4 figure