On variational and symplectic time integrators for Hamiltonian systems

Gagarina, Elena; Ambati, V.R.; Nurijanyan, S.; Vegt, J.J.W. van der; Bokhove, Onno

doi:10.1016/j.jcp.2015.11.049

Cited by 23 publications

(33 citation statements)

References 13 publications

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“…]] and averages {{.}} β α for a time-dependent function d(t) must be introduced (Gagarina et al, 2016):…”

Section: Time Discretization: Second-order Störmer-verlet Schemementioning

confidence: 99%

“…Gagarina et al (2016) showed that to obtain a second-order Störmer-Verlet scheme, P and Q must be discretized with mid-point and trapezoidal rules, respectively, that is,…”

Section: Time Discretization: Second-order Störmer-verlet Schemementioning

confidence: 99%

“…Setting P n = αP n,+ + βP n,− with α = 0 and β = 1 ensures stability of the numerical scheme (Gagarina et al, 2016).…”

Section: Time Discretization: Second-order Störmer-verlet Schemementioning

confidence: 99%

“…The present approaches are necessary to determine how, in future work, we can impose the line solitons on the wave makers to generate a 4-fold amplified wave in the middle of a wave basin and measure its impact on offshore structures. The variational technique used in the present approach enables us to express the equations as a Hamiltonian system to which robust time integrators can be applied (Hairer et al, 2006;Gagarina et al, 2016). The space and time Galerkin finite-element method used to discretize the present model ensures the overall conservation of mass, energy and momentum, which are essential in the highamplitude and long-distance propagating waves studied here.…”

Section: Introductionmentioning

confidence: 99%

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Variational modelling of extreme waves through oblique interaction of solitary waves: application to Mach reflection

Gidel

Bokhove

Kalogirou

2017

Nonlin. Processes Geophys.

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Abstract. In this work, we model extreme waves that occur due to Mach reflection through the intersection of two obliquely incident solitary waves. For a given range of incident angles and amplitudes, the Mach stem wave grows linearly in length and amplitude, reaching up to 4 times the amplitude of the incident waves. A variational approach is used to derive the bidirectional Benney-Luke equations, an asymptotic equivalent of the three-dimensional potentialflow equations modelling water waves. This nonlinear and weakly dispersive model has the advantage of allowing wave propagation in two horizontal directions, which is not the case with the unidirectional Kadomtsev-Petviashvili (KP) equation used in most previous studies. A variational Galerkin finite-element method is applied to solve the system numerically in Firedrake with a second-order Störmer-Verlet temporal integration scheme, in order to obtain stable simulations that conserve the overall mass and energy of the system. Using this approach, we are able to get close to the 4-fold amplitude amplification predicted by Miles.

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“…]] and averages {{.}} β α for a time-dependent function d(t) must be introduced (Gagarina et al, 2016):…”

Section: Time Discretization: Second-order Störmer-verlet Schemementioning

confidence: 99%

“…Gagarina et al (2016) showed that to obtain a second-order Störmer-Verlet scheme, P and Q must be discretized with mid-point and trapezoidal rules, respectively, that is,…”

Section: Time Discretization: Second-order Störmer-verlet Schemementioning

confidence: 99%

“…Setting P n = αP n,+ + βP n,− with α = 0 and β = 1 ensures stability of the numerical scheme (Gagarina et al, 2016).…”

Section: Time Discretization: Second-order Störmer-verlet Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Variational modelling of extreme waves through oblique interaction of solitary waves: application to Mach reflection

Gidel

Bokhove

Kalogirou

2017

Nonlin. Processes Geophys.

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show abstract

“…This ensures stability and overall energy conservation. Variation of this algebraic VP yields the so-called Galerkin finite-element model, with mixed dis/continuous element approximations [8]. The FSI is embedded in a single variational formulation with the associated conservative properties akin to the ones in the parent continuum system.…”

Section: T)mentioning

confidence: 99%

Variational modelling of wave–structure interactions with an offshore wind-turbine mast

2017

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We consider the development of a mathematical model of water waves interacting with the mast of an offshore wind turbine. A variational approach is used for which the starting point is an action functional describing a dual system comprising a potential-flow fluid, a solid structure modelled with nonlinear elasticity, and the coupling between them. We develop a linearized model of the fluid-structure or wave-mast coupling, which is a linearization of the variational principle for the fully coupled nonlinear model. Our numerical results for the linear case indicate that our variational approach yields a stable numerical discretization of a fully coupled model of water waves and an elastic beam. The energy exchange between the subsystems is seen to be in balance, yielding a total energy that shows only small and bounded oscillations amplitude of which tends to zero with the second-order convergence as the timestep approaches zero. Similar second-order convergence is observed for spatial mesh refinement. The linearized model so far developed can be extended to a nonlinear regime.

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Variational Water-Wave Modeling: From Deep Water to Beaches

Bokhove

2022

Mathematics of Planet Earth

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On variational and symplectic time integrators for Hamiltonian systems

Cited by 23 publications

References 13 publications

Variational modelling of extreme waves through oblique interaction of solitary waves: application to Mach reflection

Variational modelling of extreme waves through oblique interaction of solitary waves: application to Mach reflection

Variational modelling of wave–structure interactions with an offshore wind-turbine mast

Variational Water-Wave Modeling: From Deep Water to Beaches

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