Geometric exponential integrators

Shen, Xuefeng; Leok, Melvin

doi:10.1016/j.jcp.2019.01.005

Cited by 30 publications

(12 citation statements)

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“…Remark 3.2. In [41], the authors also studied the exponential integrators for the single Hamiltonian system (3.8) and proved the energy conservation. However, the proof was made under the assumption that S and L commute.…”

Section: Energy Conservation Of the Epavf Methodsmentioning

confidence: 99%

“…However, the proof was made under the assumption that S and L commute. For the nonlinear Schrödinger equation and KdV equation considered in [41], this assumption does satisfy. But for a wide class of Hamiltonian systems (e.g., the KGS and KGZ equations), this assumption is not essential.…”

Section: Energy Conservation Of the Epavf Methodsmentioning

confidence: 99%

“…This method is then extended to arbitrarily high order according to the modifying integrator theory [36]. Based on the dG method, Shen and Leok also proposed EP-EI as one kind of geometric exponential integrators studied in [41]. For particular quadratic invariants, combining the Runge-Kutta method, Bhatt and Moore [5] developed higher-order exponential integrators for linearly damped systems.…”

Section: Introductionmentioning

confidence: 99%

“…Besides the computational efficiency, we also provide a new approach to prove the energy conservation of EPAVF. In contrast to the proof techniques used in [32] and [41] where extra assumptions on S i and L i are required, our approach only utilize the symmetry properties of S i and L i which is essential for problems (1.1). Although the proof is carried out for the multi-components Hamiltonian systems (1.1), it can also be employed to re-analyze EAVF in [32] and the exponential integrators in [41].…”

Section: Introductionmentioning

confidence: 99%

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Efficient energy-preserving exponential integrators for multi-components Hamiltonian systems

Gu¹,

Jiang²,

Wang³

et al. 2021

Preprint

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In this paper, we develop a framework to construct energy-preserving methods for multi-components Hamiltonian systems, combining the exponential integrator and the partitioned averaged vector field method. This leads to numerical schemes with both advantages of long-time stability and excellent behavior for highly oscillatory or stiff problems. Compared to the existing energy-preserving exponential integrators (EP-EI) in practical implementation, our proposed methods are much efficient which can at least be computed by subsystem instead of handling a nonlinear coupling system at a time. Moreover, for most cases, such as the Klein-Gordon-Schrödinger equations and the Klein-Gordon-Zakharov equations considered in this paper, the computational cost can be further reduced. Specifically, one part of the derived schemes is totally explicit, and the other is linearly implicit. In addition, we present rigorous proof of conserving the original energy of Hamiltonian systems, in which an alternative technique is utilized so that no additional assumptions are required, in contrast to the proof strategies used for the existing EP-EI. Numerical experiments are provided to demonstrate the significant advantages in accuracy, computational efficiency, and the ability to capture highly oscillatory solutions.

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Section: Energy Conservation Of the Epavf Methodsmentioning

confidence: 99%

Section: Energy Conservation Of the Epavf Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Efficient energy-preserving exponential integrators for multi-components Hamiltonian systems

Gu¹,

Jiang²,

Wang³

et al. 2021

Preprint

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“…For an accurate solution, one should consider either the IMEX method (see [21]) or Filon-type methods (see [7]). The combination of exponential type integrators with symplectic and energy-preserving integrators was also recently considered in [18]. 5.7.…”

Section: Sv Applied To Kepler's Planar 2-body Problemmentioning

confidence: 99%

Lagrangian and Hamiltonian Taylor variational integrators

2017

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Abstract. In this paper, we present a variational integrator that is based on an approximation of the Euler-Lagrange boundary-value problem via Taylor's method. This can viewed as a special case of the shooting-based variational integrator introduced in [11]. The Taylor variational integrator exploits the structure of the Taylor method, which results in a shooting method that is one order higher compared to other shooting methods based on a one-step method of the same order. In addition, this method can generate quadrature nodal evaluations at the cost of a polynomial evaluation, which may increase its efficiency relative to other shooting-based variational integrators. A symmetric version of the method is proposed, and numerical experiments are conducted to exhibit the efficacy and efficiency of the method.

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