Diagonally implicit symplectic Runge-Kutta methods with special properties

Kalogiratou, Z.; Monovasilis, Th.; Simos, Theodore E.

doi:10.1063/1.4756416

Cited by 9 publications

(10 citation statements)

References 30 publications

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“… Method M546 : fourth-order symmetric DIRK methods for periodic stiff problems of Franco and Gómez (see [ 2 ]), a symmetric diagonally implicit Runge-Kutta method with five stages of algebraic order 4 dispersion order 6 was proposed. Method M756 : diagonally implicit symplectic Runge-Kutta methods with special properties of Kalogiratou et al are a seven-stage method with algebraic order 5 and dispersion order 6 (see [ 8 ]). Method M748 : diagonally implicit symplectic Runge-Kutta methods with special properties of Kalogiratou et al are a seven-stage method with algebraic order 4 and dispersion order 8 (see [ 8 ]).…”

Section: Stability and Dispersive Error Analysismentioning

confidence: 99%

“…The phase-lag (or dispersion) property was introduced by Brusa and Nigro [ 13 ]. In the past few years, lots of work have been done in the construction symplectic methods for oscillating problems (see [ 6 – 8 , 14 – 18 ]).…”

Section: Introductionmentioning

confidence: 99%

“…The special symplectic methods ( 4 ) have been discussed by Suris [ 14 ], Qin and Zhang [ 10 ], Kalogiratou et al [ 7 , 8 ], Cong and Jiang [ 1 ], and Franco and Gómez [ 2 ]. In [ 14 ], method was derived with order p = 3.…”

Section: Introductionmentioning

confidence: 99%

“…In [ 2 ], five-stage symmetric method with algebraic order 4 and dispersion order 6 was presented. In [ 8 ], seven-stage method with algebraic order 5 dispersion order 6 and seven-stage method with algebraic order 4 dispersion order 8 were presented. In [ 1 ], method was derived with algebraic order 6.…”

Section: Introductionmentioning

confidence: 99%

“…In Section 4 , the stability and dispersive character of the proposed method are studied. In Section 5 , numerical results are given to investigate the Hamiltonian quantity of the proposed method; the proposed method has been compared with the methods of Franco and Gómez [ 2 ], the methods of Kalogiratou et al [ 8 ], and the method of Cong and Jiang [ 1 ], and they are all Runge-Kutta methods of the format ( 4 ); from the numerical experiments, the proposed method shows some superiority.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Diagonally Implicit Symplectic Runge-Kutta Methods with High Algebraic and Dispersion Order

Yang

Cheng-xiang

2014

The Scientific World Journal

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The numerical integration of Hamiltonian systems with oscillating solutions is considered in this paper. A diagonally implicit symplectic nine-stages Runge-Kutta method with algebraic order 6 and dispersion order 8 is presented. Numerical experiments with some Hamiltonian oscillatory problems are presented to show the proposed method is as competitive as the existing same type Runge-Kutta methods.

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Section: Stability and Dispersive Error Analysismentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Diagonally Implicit Symplectic Runge-Kutta Methods with High Algebraic and Dispersion Order

Yang

Cheng-xiang

2014

The Scientific World Journal

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Variational symplectic diagonally implicit Runge-Kutta methods for isospectral systems

Silva

Lessig

2022

Bit Numer Math

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Isospectral flows appear in a variety of applications, e.g. the Toda lattice in solid state physics or in discrete models for two-dimensional hydrodynamics. The isospectral property thereby often corresponds to mathematically or physically important conservation laws. The most prominent feature of these systems, i.e. the conservation of the eigenvalues of the matrix state variable, should therefore be retained when discretizing these systems. Recently, it was shown how isospectral Runge-Kutta methods can in the Lie-Poisson case be obtained through Hamiltonian reduction of symplectic Runge-Kutta methods on the cotangent bundle of a Lie group. We provide the Lagrangian analogue and, in the case of symplectic diagonally implicit Runge-Kutta methods, derive the methods through a discrete Euler-Poincaré reduction. Our derivation relies on a formulation of diagonally implicit isospectral Runge-Kutta methods in terms of the Cayley transform, generalizing earlier work that showed this for the implicit midpoint rule. Our work is also a generalization of earlier variational Lie group integrators that, interestingly, appear when these are interpreted as update equations for intermediate time points. From a practical point of view, our results allow for a simple implementation of higher order isospectral methods and we demonstrate this with numerical experiments where both the isospectral property and energy are conserved to high accuracy.

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Mass and energy conservative high-order diagonally implicit Runge–Kutta schemes for nonlinear Schrödinger equation

Liu,

Zhang,

Qian

et al. 2024

Applied Mathematics Letters

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Diagonally implicit symplectic Runge-Kutta methods with special properties

Cited by 9 publications

References 30 publications

Diagonally Implicit Symplectic Runge-Kutta Methods with High Algebraic and Dispersion Order

Diagonally Implicit Symplectic Runge-Kutta Methods with High Algebraic and Dispersion Order

Variational symplectic diagonally implicit Runge-Kutta methods for isospectral systems

Mass and energy conservative high-order diagonally implicit Runge–Kutta schemes for nonlinear Schrödinger equation

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