Geometric Numerical Integration

Jahnke, Tobias

doi:10.1007/3-540-30666-8

Cited by 129 publications

(160 citation statements)

References 5 publications

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“…(19) and (20) with the potential variation between the electrode surface and the axis taken into account. Thus, the rms uncertainty in the measurement of the longitudinal energy, W ∥e , is …”

Section: Discussionmentioning

confidence: 99%

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2-D energy analyzer for low energy electrons

Karkare

Cultrera

Hwang

et al. 2015

Review of Scientific Instruments

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A 2-D electron energy analyzer is designed and constructed to measure the transverse and longitudinal energy distribution of low energy (<1 eV) electrons. The analyzer operates on the principle of adiabatic invariance and motion of low energy electrons in a strong longitudinal magnetic field. The operation of the analyzer is studied in detail and a design to optimize the energy resolution, signal to noise ratio, and physical size is presented. An energy resolution better than 6 meV has been demonstrated. Such an analyzer is a powerful tool to study the process of photoemission which limits the beam quality in modern accelerators.

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Section: Discussionmentioning

confidence: 99%

“…18 The trajectory of the electron in these realistic field maps was obtained by integrating the equations of motion using an 8-stage symplectic implicit integrator. 19 Figure 1 shows the trajectory of an electron obtained by such a simulation.…”

Section: Verification By Simulationmentioning

confidence: 99%

2-D energy analyzer for low energy electrons

Karkare

Cultrera

Hwang

et al. 2015

Review of Scientific Instruments

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“…Since we are applying a symplectic method ψ to a Hamiltonian ODEż = f (z) in extended phase space with Hamiltonian K, we can apply the result from [16,Theorem IX.3.1], showing that the modified equation is also Hamiltonian, and has Hamiltoniañ…”

Section: Corollary 2 Symplectic Partitioned Runge-kutta (Sprk) Methodmentioning

confidence: 99%

“…The idea of devising numerical methods which are themselves symplectic maps goes back into the previous century, some early references are [9,27]. The monographs by Hairer et al [16], and Leimkuhler and Reich [22] may be consulted for an extensive treatment. Such numerical methods are called symplectic integrators and their success is often explained through the well known fact that any symplectic map can be identified as the exact flow of a local, perturbed Hamiltonian problem.…”

Section: Introductionmentioning

confidence: 98%

Geometric integration of non-autonomous linear Hamiltonian problems

Marthinsen

Owren

2015

Adv Comput Math

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Symplectic integration of autonomous Hamiltonian systems is a wellknown field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of dimensions. We show that one possible extension of symplectic methods in the autonomous setting to the non-autonomous setting is obtained by using canonical transformations. Many existing methods fit into this framework. We also perform experiments which indicate that for exponential integrators, the canonical and symmetric properties are important for good long time behaviour. In particular, the theoretical and numerical results support the well documented fact from the literature that exponential integrators for non-autonomous linear problems have superior accuracy compared to general ODE schemes.

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“…To circumvent the limitations of standard time integration schemes, several general methods for conservative or dissipative ODEs such as orthogonal projection [29, Section IV.4] and relaxation [33,52,55] as well as more problem-dependent methods for dissipative ODEs such as artificial dissipation or filtering [28,40,63] have been proposed, mainly in the context of one-step methods. For fluid mechanics applications, orthogonal projection methods are not really suitable since they do not preserve linear invariants such as the total mass, [29]. In contrast, relaxation methods conserve all linear invariants and can still preserve the correct global entropy conservation/dissipation in time [33,51,52,55].…”

Section: Related Workmentioning

confidence: 99%

Relaxation Runge--Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier--Stokes Equations

Ranocha¹,

Sayyari²,

Dalcín³

et al. 2020

SIAM J. Sci. Comput.

113

118

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Recently, relaxation methods have been developed to guarantee the preservation of a single global functional of the solution of an ordinary differential equation We generalize this approach to guarantee local entropy inequalities for finitely many convex functionals (entropies) and apply the resulting methods to the compressible Euler and Navier-Stokes equations. Based on the unstructured hp-adaptive SSDC framework of entropy conservative or dissipative semidiscretizations using summation-by-parts and simultaneous-approximation-term operators, we develop the first discretizations for compressible computational fluid dynamics that are primary conservative, locally entropy stable in the fully discrete sense under a usual CFL condition, explicit except for the parallelizable solution of a single scalar equation per element, and arbitrarily high-order accurate in space and time. We demonstrate the accuracy and the robustness of the fully-discrete explicit locally entropy-stable solver for a set of test cases of increasing complexity.

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Geometric Numerical Integration

Abstract: The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Cited by 129 publications

References 5 publications

2-D energy analyzer for low energy electrons

2-D energy analyzer for low energy electrons

Geometric integration of non-autonomous linear Hamiltonian problems

Relaxation Runge--Kutta Methods: Fully Discrete Explicit Entropy-Stable Schemes for the Compressible Euler and Navier--Stokes Equations

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