Explicit high-order energy-preserving exponential time differencing method for nonlinear Hamiltonian PDEs

Xu, Zhuangzhi; Cai, Wenjun; Song, Yisheng; Wang, Yushun

doi:10.1016/j.amc.2021.126208

Cited by 5 publications

(2 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, the implicit middle scheme can preserve the quadratic energy conservation. Wang et al proposed explicit energy conserving schemes for the reformulated system of the Hamiltonian system by the project method [13,29]. Song et al proposed the explicit energy conserving method of the Hamiltonian system based on the relaxation Runge-Kutta method [33].…”

Section: Introductionmentioning

confidence: 99%

High order explicit energy preserving Lie group method for some Hamiltonian systems

Sun

2023

Preprint

View full text Add to dashboard Cite

By the nonlinear transformation, some classical Hamiltonian systems can be written as the reformulation systems, which have the quadratic positive definite energy conserving function. These new reformulated systems are transformed into the ordinary differential equations on manifolds by the linear transformation. The explicit Runge-Kutta Munthe-Kaas (RKMK) method, which is a kind of Lie group method, is applied to solve the differential equations on manifolds. Therefore, a explicit energy preserving(EEP) RKMK method is proposed. The EEP-RKMK method is applied to solve the two classical dynamical systems: the pendulum problem and the Hénon-Heiles system. Thus, the explicit energy conserving schemes of the two classical dynamical systems are obtained. Numerical simulations investigate the effective of these new schemes in preserving energy conserving property of these equations and well simulating the dynamical behaviors. AMS subject classification: 37K05 65M20 65M70

show abstract

Section: Introductionmentioning

confidence: 99%

High order explicit energy preserving Lie group method for some Hamiltonian systems

Sun

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Combinations of exponential integrators and the SAV approach have also been investigated [23][24][25][26]. In particular, Jiang, Cui, Qian, and Song [27] propose highorder linearly implicit structure-preserving exponential integrators for the nonlinear Schrödinger equation based on the SAV approach and the Lawson transformation [28].…”

Section: Introductionmentioning

confidence: 99%

High-order linearly implicit exponential integrators conserving quadratic invariants with application to scalar auxiliary variable approach

Sato

2023

Preprint

View full text Add to dashboard Cite

This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical instances and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato, Miyatake, and Butcher (Appl. Numer. Math., 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes. MSC Classification: 65L05 , 65M06 , 65P10

show abstract

High-order linearly implicit exponential integrators conserving quadratic invariants with application to scalar auxiliary variable approach

Sato

2024

Numer Algor

View full text Add to dashboard Cite

This paper proposes a framework for constructing high-order linearly implicit exponential integrators that conserve a quadratic invariant. This is then applied to the scalar auxiliary variable (SAV) approach. Quadratic invariants are significant objects that are present in various physical equations and also in computationally efficient conservative schemes for general invariants. For instance, the SAV approach converts the invariant into a quadratic form by introducing scalar auxiliary variables, which have been intensively studied in recent years. In this vein, Sato et al. (Appl. Numer. Math. 187, 71-88 2023) proposed high-order linearly implicit schemes that conserve a quadratic invariant. In this study, it is shown that their method can be effectively merged with the Lawson transformation, a technique commonly utilized in the construction of exponential integrators. It is also demonstrated that combining the constructed exponential integrators and the SAV approach yields schemes that are computationally less expensive. Specifically, the main part of the computational cost is the product of several matrix exponentials and vectors, which are parallelizable. Moreover, we conduct some mathematical analyses on the proposed schemes.

show abstract

Explicit high-order energy-preserving exponential time differencing method for nonlinear Hamiltonian PDEs

Cited by 5 publications

References 35 publications

High order explicit energy preserving Lie group method for some Hamiltonian systems

High order explicit energy preserving Lie group method for some Hamiltonian systems

High-order linearly implicit exponential integrators conserving quadratic invariants with application to scalar auxiliary variable approach

High-order linearly implicit exponential integrators conserving quadratic invariants with application to scalar auxiliary variable approach

Contact Info

Product

Resources

About