A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type

Crank, John P.; Nicolson, P.; Hartree, D. R.

doi:10.1017/s0305004100023197

Cited by 2,648 publications

(659 citation statements)

References 6 publications

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“…ν is the viscosity of the gas, which we parametrize such that the viscous time-scale is related to the dynamical time-scale via t visc = ξ t dyn with a constant parameter ξ of order 10 2 -10 3 (Duschl, Strittmatter & Biermann 2000). 1 We carry out the time integration of equation (3) by applying an implicit Crank-Nicolson finite differences scheme (Crank, Nicolson & Hartree 1947). The radial calculation domain extends from an inner radius s in to the disc's initial outer radius s out, 0 .…”

Section: Evolution Of the Agnmentioning

confidence: 99%

Viscous time lags between starburst and AGN activity

Blank

Duschl

2016

Mon. Not. R. Astron. Soc.

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There is strong observational evidence indicating a time lag of order of some 100 Myr between the onset of starburst and AGN activity in galaxies. Dynamical time lags have been invoked to explain this. We extend this approach by introducing a viscous time lag the gas additionally needs to flow through the AGN's accretion disc before it reaches the central black hole. Our calculations reproduce the observed time lags and are in accordance with the observed correlation between black hole mass and stellar velocity dispersion.Key words: galaxies: active -galaxies: formation -galaxies: interactions -galaxies: nucleiquasars: general -galaxies: starburst. I N T RO D U C T I O NMotivated by, for instance, observed correlations between the mass of an AGN's central black hole and the host galaxy's velocity dispersion (e.g. Gebhardt et al. 2000) and between black hole mass and bulge mass (e.g. Kormendy & Richstone 1995), there is an ongoing debate whether, and if so, how starbursts and AGN are connected to each other.Di , for instance, explain such correlations as due to a thermal AGN feedback that heats the gas of the galaxy and thus prevents further star formation and AGN activity: more massive galaxies have a deeper gravitational potential well, thus the black hole has to gain more mass before its luminosity is capable of expelling the gas from the galaxy and quenching star formation and AGN activity. This then leads to the velocity dispersion and the bulge mass, respectively, to be related to the black hole mass. In these simulations starburst and AGN activity occur simultaneously, but recent observations show that AGN activity may be delayed with regard to star formation activity by time-scales of 50-250 Myr (e.g. Davies et al. 2007;Schawinski et al. 2009;Wild, Heckman & Charlot 2010).Hopkins (2012) argues that such a time lag can occur for purely dynamical reasons. His high spatial resolution simulations of galaxy mergers show first an inward motion of gas towards the dynamical centre giving rise to (a burst of) star formation. In these models, the gas flowing further inwards can do so only by losing angular momentum by gravitational instabilities. This, in turn, gives rise to a time lag between star formation and AGN activity.We extend this idea by modelling the loss of angular momentum and the ensuing inflow in the framework of an accretion disc scenario. Thus the time lag between starburst and AGN activity consists of a dynamical lag given by the time span the gas needs to E-mail: mblank@astrophysik.uni-kiel.de reach the accretion disc and a subsequent viscous lag given by the time span the gas needs to flow through the accretion disc until it reaches the black hole.In Section 2 we explain our numerical methods and the setup of the merger event that is, in this scenario, responsible for the inflow of gas to the newly forming galactic centre. In Section 3 we present and discuss the general picture that results from our calculations. As our model depends on a number of parameters, we perform a parameter stud...

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Section: Evolution Of the Agnmentioning

confidence: 99%

Viscous time lags between starburst and AGN activity

Blank

Duschl

2016

Mon. Not. R. Astron. Soc.

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“…[12] We solve the diffusion equation (1) assuming a discrete meshed grid of dimension N (typically 91 cells) from 1 < L* < 10 and use the Crank-Nicolson scheme [Crank and Nicolson, 1947], which is an implicit, numerically stable method that does not need to satisfy the Courant condition [Press et al, 1986]. We use a parameterized form of the diffusion coefficient that is a function of magnetic activity [Brautigam and Albert, 2000] …”

Section: Radial Diffusion Modelmentioning

confidence: 99%

Identifying the radiation belt source region by data assimilation

et al. 2007

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[1] We describe how assimilation of radiation belt data with a simple radial diffusion code can be used to identify and adjust for unknown physics in the model. We study the dropout and the following enhancement of relativistic electrons during a moderate storm on 25 October 2002. We introduce a technique that uses an ensemble Kalman filter and the probability distribution of the forecast ensemble to identify if the model is drifting away from the observations and to find inconsistencies between model forecast and observations. We use the method to pinpoint the time periods and locations where most of the disagreement occurs and how much the Kalman filter has to adjust the model state to match the observations. Although the model does not contain explicit source or loss terms, the Kalman filter algorithm can implicitly add very localized sources or losses in order to reduce the discrepancy between model and observations. We use this technique with multisatellite observations to determine when simple radial diffusion is inconsistent with the observed phase space densities indicating where additional source (acceleration) or loss (precipitation) processes must be active. We find that the outer boundary estimated by the ensemble Kalman filter is consistent with negative phase space density gradients in the outer electron radiation belt. We also identify specific regions in the radiation belts (L* % 5-6 and to a minor extend also L* % 4) where simple radial diffusion fails to adequately capture the variability of the observations, suggesting local acceleration/loss mechanisms.

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“…Equation (1) was solved numerically using the CrankNicolson method 2 and gating variables were computed using the Euler method. 3 In both models, a time step of 15 µs and a spatial step of 100 µm were used.…”

Section: Modelmentioning

confidence: 99%

Reproducing Cardiac Restitution Properties Using the Fenton–Karma Membrane Model

Oliver

Krassowska

2005

Ann Biomed Eng

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Abstract-Modern models of cardiac membranes are often highly complex and computationally expensive, particularly when used in long simulations of spatially extended models of cardiac tissue. Therefore, there is a need for simpler membrane models that preserve the features of the complex models deemed important. This communication describes an empirical procedure that was used to choose the parameters of the three-variable FentonKarma (FK3V) model to reproduce the restitution properties of the Courtemanche-Ramirez-Nattel model of atrial tissue. The resulting parameter values for the FK3V model and the sensitivity table for all its parameters are provided. Thus, this study gives insight into the behavior of the FK3V model and the effect of its parameters on restitution properties.

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A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type

Cited by 2,648 publications

References 6 publications

Viscous time lags between starburst and AGN activity

Viscous time lags between starburst and AGN activity

Identifying the radiation belt source region by data assimilation

Reproducing Cardiac Restitution Properties Using the Fenton–Karma Membrane Model

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