Fehlerabsch�tzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus

Bulirsch, Roland; Stoer, Josef

doi:10.1007/bf01386092

Cited by 302 publications

(190 citation statements)

References 8 publications

(2 reference statements)

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“…We have extrapolated the results of the RG calculations using the BST-algoritm [27], which is known to be a good tool to extrapolate finite lattice data [28]. The results are also included in table 1.…”

Section: Resultsmentioning

confidence: 99%

One-dimensional contact process: Duality and renormalization

Hooyberghs

Vanderzande

2001

Phys. Rev. E

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We study the one-dimensional contact process in its quantum version using a recently proposed real space renormalisation technique for stochastic many-particle systems. Exploiting the duality and other properties of the model, we can apply the method for cells with up to 37 sites. After suitable extrapolation, we obtain exponent estimates which are comparable in accuracy with the best known in the literature.

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Section: Resultsmentioning

confidence: 99%

One-dimensional contact process: Duality and renormalization

Hooyberghs

Vanderzande

2001

Phys. Rev. E

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“…For Q < 1, however, because no zero lies on the unit circle for a finite-size lattice, we need to calculate the FM critical point from the zero closest to the point p c = 1 (the first zero). By using the Bulirsch-Stoer (BST) algorithm [6] we extrapolated our results for finite lattices to infinite size. The error estimates are twice the difference between the (n − 1, 1) and (n − 1, 2) approximants.…”

Section: The Ferromagnetic Potts Modelmentioning

confidence: 99%

Partition function zeros of the Q-state Potts model for non-integer Q

Kim¹,

Creswick²,

Chen³

et al. 2000

Physica A: Statistical Mechanics and its Applications

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The distribution of the zeros of the partition function in the complex temperature plane (Fisher zeros) of the two-dimensional Q-state Potts model is studied for non-integer Q. On L × L self-dual lattices studied (L ≤ 8), no Fisher zero lies on the unit circle p 0 = e iθ in the complex p = (e βJ − 1)/ √ Q plane for Q < 1, while some of the Fisher zeros lie on the unit circle for Q > 1 and the number of such zeros increases with increasing Q. The ferromagnetic and antiferromagnetic properties of the Potts model are investigated using the distribution of the Fisher zeros. For the Potts ferromagnet we verify the den Nijs formula for the thermal exponent y t . For the Potts antiferromagnet we also verify the Baxter conjecture for the critical temperature and present new results for the thermal exponents in the range 0 < Q < 3.

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“…To handle the N-body calculations accurately we implemented a Bulirsch -Stoer integrator (Bulirsch & Stoer 1964) parallelised to run on the GPU and thus gaining a speed factor of up to 40.…”

Section: Code Descriptionmentioning

confidence: 99%

Disc-protoplanet interaction

Gyergyovits

Eggl

Pilat-Lohinger

et al. 2014

A&A

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Context. More than 60 planets have been discovered so far in systems that harbour two stars, some of which have binary semi-major axes as small as 20 au. It is well known that the formation of planets in such systems is strongly influenced by the stellar components, since the protoplanetary disc and the particles within are exposed to the gravitational influence of the binary. However, the question on how self-gravitating protoplanetary bodies affect the evolution of a radiative, circumprimary disc is still open. Aims. We present our 2D hydrodynamical GPU-CPU code and study the interaction of several thousands of self-gravitating particles with a viscous and radiative circumprimary disc within a binary star system. To our knowledge this program is the only one at the moment that is capable to handle this many particles and to calculate their influence on each other and on the disc. Methods. We performed hydrodynamical simulations of a circumstellar disc assuming the binary system to be coplanar. Our gridbased staggered mesh code relies on ideas from ZEUS-2D, where we implemented the FARGO algorithm and an additional energy equation for the radiative cooling according to opacity tables. To treat particle motion we used a parallelised version of the precise Bulirsch -Stoer algorithm. Four models in total where computed taking into account (i) only N-body interaction; (ii) N-body and disc interaction; (iii) the influence of computational parameters (especially smoothing) on N-body interaction; and (iv) the influence of a quiet low-eccentricity disc while running model (ii). The impact velocities were measured at two different time intervals and were compared. Results. We show that the combination of disc-and N-body self-gravity can have a significant influence on the orbit evolution of roughly Moon sized protoplanets. Conclusions. Not only gas drag can alter the orbit of particles, but the gravitational influence of the disc can accomplish this as well. The results depend strongly on the state of the disc (i.e. quiet or dynamically evolving) -according to encounter-probability distributions, planet formation can be strongly altered if there is a dynamically evolving gas disc -and also on the smoothing parameter.

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Fehlerabsch�tzungen und Extrapolation mit rationalen Funktionen bei Verfahren vom Richardson-Typus

Abstract: Sei T(h) die zu einem Diskretisierungsparameter h gewonnene numerische N/iherung eines exakten Problems, definiert durch lim T(h)= T(0).

Cited by 302 publications

References 8 publications

One-dimensional contact process: Duality and renormalization

One-dimensional contact process: Duality and renormalization

Partition function zeros of the Q-state Potts model for non-integer Q

Disc-protoplanet interaction

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