arXiv subjects

Xinyuan Wu

Publications and source records attributed to Xinyuan Wu.

27 records · Page 2Linked to original sources

arXiv2018

Exponential collocation methods for the cubic Schrödinger equation

In this paper we derive and analyse new exponential collocation methods to efficiently solve the cubic Schrödinger Cauchy problem on a $d$-dimensional torus. Energy preservation is a key feature of the cubic Schrödinger equation. It is proved that the novel methods can be of arbitrarily high order which exactly or nearly preserve the continuous energy of the original continuous system. The existence and uniqueness, regularity, global convergence, nonlinear stability of the new methods are studied in detail. Two practical exponential collocation methods are constructed and two numerical experiments are included. The numerical results illustrate the efficiency of the new methods in comparison with existing numerical methods in the literature.

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arXiv2018

Long-time momentum and actions behaviour of energy-preserving methods for semilinear wave equations via spatial spectral semi-discretizations

As is known that wave equations have physically very important properties which should be respected by numerical schemes in order to predict correctly the solution over a long time period. In this paper, the long-time behaviour of momentum and actions for energy-preserving methods is analysed for semilinear wave equations. A full discretisation of wave equations is derived and analysed by firstly using a spectral semi-discretisation in space and then by applying the adopted average vector field (AAVF) method in time. This numerical scheme can exactly preserve the energy of the semi-discrete system. The main theme of this paper is to analyse another important physical property of the scheme. It is shown that this scheme yields near conservation of a modified momentum and modified actions over long times. Both the results are rigorously proved based on the technique of modulated Fourier expansions in two stages. First a multi-frequency modulated Fourier expansion of the AAVF method is constructed and then two almost-invariants of the modulation system are derived.

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arXiv2018

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.

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arXiv2018

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--Nyström (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.

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arXiv2018

Volume-preserving exponential integrators

As is known that various dynamical systems including all Hamiltonian systems preserve volume in phase space. This qualitative geometrical property of the analytical solution should be respected in the sense of Geometric Integration. This paper analyses the volume-preserving property of exponential integrators in different vector fields. We derive a necessary and sufficient condition of volume preservation for exponential integrators, and with this condition, volume-preserving exponential integrators are analysed in detail for four kinds of vector fields. It turns out that symplectic exponential integrators can be volume preserving for a much larger class of vector fields than Hamiltonian systems. On the basis of the analysis, novel volume-preserving exponential integrators are derived for solving highly oscillatory second-order systems and extended Runge--Kutta--Nyström (ERKN) integrators of volume preservation are presented for separable partitioned systems. Moreover, the volume preservation of Runge--Kutta--Nyström (RKN) methods is also discussed. Four illustrative numerical experiments are carried out to demonstrate the notable superiority of volume-preserving exponential integrators in comparison with volume-preserving Runge-Kutta methods.

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arXiv2018

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--Nyström 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.

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arXiv2017

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.

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arXiv2016

Exponential Fourier collocation methods for solving first-order differential equations

In this paper, a novel class of exponential Fourier collocation methods (EFCMs) is presented for solving systems of first-order ordinary differential equations. These so-called exponential Fourier collocation methods are based on the variation-of-constants formula, incorporating a local Fourier expansion of the underlying problem with collocation methods. We discuss in detail the connections of EFCMs with trigonometric Fourier collocation methods (TFCMs), the well-known Hamiltonian Boundary Value Methods (HBVMs), Gauss methods and Radau IIA methods. It turns out that the novel EFCMs are an essential extension of these existing methods. We also analyse the accuracy in preserving the quadratic invariants and the Hamiltonian energy when the underlying system is a Hamiltonian system. Other properties of EFCMs including the order of approximations and the convergence of fixed-point iterations are investigated as well. The analysis given in this paper proves further that EFCMs can achieve arbitrarily high order in a routine manner which allows us to construct higher-order methods for solving systems of first-order ordinary differential equations conveniently. We also derive a practical fourth-order EFCM denoted by EFCM(2,2) as an illustrative example. The numerical experiments using EFCM(2,2) are implemented in comparison with an existing fourth-order HBVM, an energy-preserving collocation method and a fourth-order exponential integrator in the literature. The numerical results demonstrate the remarkable efficiency and robustness of the novel EFCM(2,2).

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arXiv2016

Trigonometric collocation methods based on Lagrange basis polynomials for multi-frequency oscillatory second-order differential equations

In the present work, a kind of trigonometric collocation methods based on Lagrange basis polynomials is developed for effectively solving multi-frequency oscillatory second-order differential equations $q^{\prime\prime}(t)+Mq(t)=f\big(q(t)\big)$. The properties of the obtained methods are investigated. It is shown that the convergent condition of these methods is independent of $\norm{M}$, which is very crucial for solving oscillatory systems. A fourth-order scheme of the methods is presented. Numerical experiments are implemented to show the remarkable efficiency of the methods proposed in this paper.

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