Comparison of some Lie-symmetry-based integrators

Chhay, Marx; Hoarau, Emma; Hamdouni, Aziz; Sagaut, Pierre

doi:10.1016/j.jcp.2010.12.015

Cited by 25 publications

(41 citation statements)

References 40 publications

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“…However, combining these invariants into a numerically stable scheme is rather complicated. Instead, we follow the idea in [79][80][81], which consists in modifying classical schemes so as to make them invariant under the symmetry group. To show how to do this, we need to formalize the notion of a discretization.…”

Section: E(g(z U)) = E(z U) E(z U)mentioning

confidence: 99%

“…This shows the ability of invariantized scheme to respect the physics of the equation and the importance of preserving symmetries at discrete scales. Other numerical tests are presented in [74,81,170]. The last geometric integrator that we shall present is the discrete exterior calculus.…”

Section: Numerical Testsmentioning

confidence: 99%

“…It is then important that numerical schemes do not break symmetries if one wishes to reproduce numerically the cited model solutions and properties. Preserving the symmetries of the equations at the discrete level is the founding key of invariant integrators [77][78][79][80][81][82][83], as will be seen in "Invariant integrators" section. Note that the symmetry structure does not exclude simplecticity structure.…”

Section: Introductionmentioning

confidence: 99%

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A review of some geometric integrators

Razafindralandy

Hamdouni

Chhay

2018

Adv. Model. and Simul. in Eng. Sci.

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Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler-Lagrange equations) and the conservation laws (Noether's theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

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Section: E(g(z U)) = E(z U) E(z U)mentioning

confidence: 99%

Section: Numerical Testsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A review of some geometric integrators

Razafindralandy

Hamdouni

Chhay

2018

Adv. Model. and Simul. in Eng. Sci.

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“…Invariant numerical schemes for Eq. (2) have already been investigated in the literature [9][10][11]21]. The schemes constructed in these references preserve the entire five-parameter symmetry group G of Burgers equation.…”

Section: Invariant Finite Difference Schemes For Burgers Equationsmentioning

confidence: 99%

“…In fact, the problem found and analyzed in [1] has been investigated quite intensively in the field of group analysis of differential and difference equations, see e.g. [2,3,10,11,14,15,19] and references therein for some of the most recent results. In particular, it was established by Dorodnitsyn and collaborators [6,11,12] that it is not possible to maintain the Galilean invariance of partial differential equations in a finite difference model when the mesh does not move in the course of the numerical integration.…”

Section: Introductionmentioning

confidence: 99%

Convecting reference frames and invariant numerical models

Bihlo

Nave

2014

Journal of Computational Physics

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In the recent paper by Bernardini et al. [J. Comput. Phys. 232 (2013), 1-6] the discrepancy in the performance of finite difference and spectral models for simulations of flows with a preferential direction of propagation was studied. In a simplified investigation carried out using the viscous Burgers equation the authors attributed the poorer numerical results of finite difference models to a violation of Galilean invariance in the discretization and propose to carry out the computations in a reference frame moving with the bulk velocity of the flow. Here we further discuss this problem and relate it to known results on invariant discretization schemes. Non-invariant and invariant finite difference discretizations of Burgers equation are proposed and compared with the discretization using the remedy proposed by Bernardini et al..

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Heterogeneous asynchronous time integrators for computational structural dynamics

Gravouil

Combescure

Brun

2014

Numerical Meth Engineering

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SUMMARYComputational structural dynamics plays an essential role in the simulation of linear and nonlinear systems. Indeed, the characteristics of the time integration procedure have a critical impact on the feasibility of the calculation. In order to go beyond the classical approach (a unique time integrator and a unique timescale), the pioneer approach of Belytschko and co-workers consisted in developing mixed implicit-explicit time integrators for structural dynamics. In a first step, the implementation and stability analyses of partitioned integrators with one time step have been achieved for a large class of time integrators. In a second step, the implementation and stability analyses of partitioned integrators with different time steps were studied in detail for particular cases. However, stability results involving different time steps and different time integrators in different parts of the mesh is still an open question in the general case for structural dynamics. The aim of this paper is to propose a state-of-the art of heterogeneous (different time schemes) asynchronous (different time steps) time integrators (HATI) for computational structural dynamics. Finally, an alternative approach based on energy considerations (with velocity continuity at the interface) is proposed in order to develop a general class of HATI for structural dynamics.

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Comparison of some Lie-symmetry-based integrators

Cited by 25 publications

References 40 publications

A review of some geometric integrators

A review of some geometric integrators

Convecting reference frames and invariant numerical models

Heterogeneous asynchronous time integrators for computational structural dynamics

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