Dynamics, Numerical Analysis, and Some Geometry

Gauckler, Ludwig J.; Hairer, Ernst; Lubich, Christian

doi:10.1142/9789813272880_0020

Cited by 9 publications

(15 citation statements)

References 78 publications

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“…The main result of the present work is the following Theorem 4.1. Let us consider the discrete dynamical system generated by the numerical scheme (15), that is, the dynamical system generated by the operator…”

Section: Remark 2 (Accuracy)mentioning

confidence: 99%

“…Proof. The fact that the numerical scheme (15) has the same equilibria as the continuous system is straightforward to check.…”

Section: Remark 2 (Accuracy)mentioning

confidence: 99%

“…and taking the inner product with 1 h (u n+1 − u n ) = 1 2 (e h v n+1 + e −h v n ), this being the first equation in (15), yields the discrete energy balance…”

Section: Remark 2 (Accuracy)mentioning

confidence: 99%

“…However, provided that the step size h is small enough, if the potential E is convex and has globally Lipschitz continuous gradient one can retrace the same qualitative convergence to equilibrium as for the AVF discretization. Notably, the Lipschitz condition on the gradient of the potential is standard in convex optimization [33], but also highly restrictive as its imposes an at most quadratic growth of E at infinity To prove our claim, we start by forcing (28) into a form similar to that of (15). In this sense, observe that using second equation in ( 28)…”

mentioning

confidence: 92%

“…This, together with correctly reproducing the energy-structure of the problem and its asymptotic energy decay rate are the main objectives of this work and they relate to one of the fundamental questions of geometric numerical integration: "How can numerical methods be constructed that respect the geometry of the problem at hand?" (see [15]).…”

mentioning

confidence: 96%

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Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems

Viorel¹,

Alecsa²,

Pinţa³

2021

DCDS

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The present work deals with the numerical long-time integration of damped Hamiltonian systems. The method that we analyze combines a specific Strang splitting, that separates linear dissipative effects from conservative ones, with an energy-preserving averaged vector field (AVF) integrator for the Hamiltonian subproblem. This construction faithfully reproduces the energydissipation structure of the continuous model, its equilibrium points and its natural Lyapunov function. As a consequence of these structural similarities, both the convergence to equilibrium and, more interestingly, the energy decay rate of the continuous dynamical system are recovered at a discrete level. The possibility of replacing the implicit AVF integrator by an explicit Störmer-Verlet one is also discussed, while numerical experiments illustrate and support the theoretical findings.

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Section: Remark 2 (Accuracy)mentioning

confidence: 99%

“…Proof. The fact that the numerical scheme (15) has the same equilibria as the continuous system is straightforward to check.…”

Section: Remark 2 (Accuracy)mentioning

confidence: 99%

“…and taking the inner product with 1 h (u n+1 − u n ) = 1 2 (e h v n+1 + e −h v n ), this being the first equation in (15), yields the discrete energy balance…”

Section: Remark 2 (Accuracy)mentioning

confidence: 99%

mentioning

confidence: 92%

mentioning

confidence: 96%

See 3 more Smart Citations

Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems

Viorel¹,

Alecsa²,

Pinţa³

2021

DCDS

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Structure-Preserving Algorithms with Uniform Error Bound and Long-time Energy Conservation for Highly Oscillatory Hamiltonian Systems

Wang

Jiang

2023

J Sci Comput

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Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

Bao

Zhao

2019

Journal of Computational Physics

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Different efficient and accurate numerical methods have recently been proposed and analyzed for the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter ε ∈ (0, 1], which is inversely proportional to the speed of light. In the nonrelativestic limit regime, i.e. 0 < ε 1, the solution of the NKGE propagates waves with wavelength at O(1) and O(ε 2 ) in space and time, respectively, which brings significantly numerical burdens in designing numerical methods. We compare systematically spatial/temporal efficiency and accuracy as well as ε-resolution (or ε-scalability) of different numerical methods including finite difference time domain methods, time-splitting method, exponential wave integrator, limit integrator, multiscale time integrator, two-scale formulation method and iterative exponential integrator. Finally, we adopt the multiscale time integrator to study the convergence rates from the NKGE to its limiting models when ε → 0 + .

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Dynamics, Numerical Analysis, and Some Geometry

Cited by 9 publications

References 78 publications

Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems

Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems

Structure-Preserving Algorithms with Uniform Error Bound and Long-time Energy Conservation for Highly Oscillatory Hamiltonian Systems

Comparison of numerical methods for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

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