| NUMERICAL CODE |
Smoothed Particle Hydrodynamics:
To study the dynamical evolution of rotational instabilities, the method of
fluid modeling known as Smoothed Particle Hydrodynamics (SPH) is employed.
This technique uses the Lagrangean form of the hydrodynamics equations
(assuming Newtonian gravity) to model a fluid as a collection of N
discrete elements or particles. The particles are initially distributed
with a specified density distribution, and are evolved according to the
fluid conservation equations (Monaghan 1988; Hernquist & Katz 1989; Benz
1990).
The values of continuous variables are determined by an interpolation or
smoothing of the nearby particle distribution using a special weighting
function known as the smoothing kernel which has a given
length-scale: the smoothing length. Because the method is
gridless, resolution is controlled by the smoothing length which is a
measure of the mean interparticle spacing (Lucy 1977; Gingold & Monaghan
1977; Monaghan 1992). This type of fluid modeling is ideally suited for
expanding and contracting astrophysical systems since it locates particles
with the material and avoids calculating empty regions of space.
I am using the implementation of SPH known as TREESPH (Hernquist & Katz
1989). This code uses the Barnes & Hut (1986) hierarchical tree algorithm
to optimize the data structures. The dynamic nature of this type of
structure permits the use of individual particle smoothing lengths and time
steps, making it spatially and temporally adaptive. By using a fixed
number of smoothing neighbors, Ns=64, the same level of
accuracy is achieved in all fluid regions. The geometric rather than the
arithmetic form of the hydrodynamic equations is used to avoid the
occurrence of negative thermal energies when using the leapfrog integration
scheme (Hernquist & Katz 1989). The entropy is evolved which allows the
conservation of total energy to be an indicator of the global accuracy of
the calculation.
To dissipate the energy added to the fluid from shocks, an artificial
viscous pressure term is introduced into the momentum and energy equations.
The omission of this term in the inviscid hydrodynamics equations results
in large unphysical oscillations in the field variables in the wake of a
shock. TREESPH incorporates three "pressure-like" artificial
viscosities. We have chosen to use the viscosity which is multiplied by
the curl of the velocity field in order to reduce the viscous shear
component often accompanied with the use of such terms (Hernquist & Katz
1989; Centrella & McMillan 1993; Houser 1996).
References:
Barnes, J.E. & Hut, P. 1986, Nature, 324, No.
4, 446.
Benz, W. 1990, in Numerical Modeling of Stellar Pulsation:
Problems and Prospects, ed. J.R. Buchler (Dordrecht: Kluwer),
269.
Gingold, R.A. & Monaghan, J.J. 1977, Mon. Not. R. Astron.
Soc., 181, 375.
Hernquist, L. & Katz, N. 1989, Ap.J.Supp., 70, 419.
Houser, J.L. 1996, Ph.D. Dissertation, Drexel University.
Lucy, L.B. 1977, Astron. J., 82, 1013.
Monaghan, J.J. 1988, Comp. Phys. Comm., 48, 89.
Monaghan, J.J. 1992, Ann. Rev. Astron. Astrophys., 30,
543.
-
To study the dynamical evolution of rotational instabilities, the method of
fluid modeling known as Smoothed Particle Hydrodynamics (SPH) is employed.
This technique uses the Lagrangean form of the hydrodynamics equations
(assuming Newtonian gravity) to model a fluid as a collection of N
discrete elements or particles. The particles are initially distributed
with a specified density distribution, and are evolved according to the
fluid conservation equations (Monaghan 1988; Hernquist & Katz 1989; Benz
1990).
-
The values of continuous variables are determined by an interpolation or
smoothing of the nearby particle distribution using a special weighting
function known as the smoothing kernel which has a given
length-scale: the smoothing length. Because the method is
gridless, resolution is controlled by the smoothing length which is a
measure of the mean interparticle spacing (Lucy 1977; Gingold & Monaghan
1977; Monaghan 1992). This type of fluid modeling is ideally suited for
expanding and contracting astrophysical systems since it locates particles
with the material and avoids calculating empty regions of space.
-
I am using the implementation of SPH known as TREESPH (Hernquist & Katz
1989). This code uses the Barnes & Hut (1986) hierarchical tree algorithm
to optimize the data structures. The dynamic nature of this type of
structure permits the use of individual particle smoothing lengths and time
steps, making it spatially and temporally adaptive. By using a fixed
number of smoothing neighbors, Ns=64, the same level of
accuracy is achieved in all fluid regions. The geometric rather than the
arithmetic form of the hydrodynamic equations is used to avoid the
occurrence of negative thermal energies when using the leapfrog integration
scheme (Hernquist & Katz 1989). The entropy is evolved which allows the
conservation of total energy to be an indicator of the global accuracy of
the calculation.
-
To dissipate the energy added to the fluid from shocks, an artificial
viscous pressure term is introduced into the momentum and energy equations.
The omission of this term in the inviscid hydrodynamics equations results
in large unphysical oscillations in the field variables in the wake of a
shock. TREESPH incorporates three "pressure-like" artificial
viscosities. We have chosen to use the viscosity which is multiplied by
the curl of the velocity field in order to reduce the viscous shear
component often accompanied with the use of such terms (Hernquist & Katz
1989; Centrella & McMillan 1993; Houser 1996).
References:
Barnes, J.E. & Hut, P. 1986, Nature, 324, No.
4, 446.
Benz, W. 1990, in Numerical Modeling of Stellar Pulsation:
Problems and Prospects, ed. J.R. Buchler (Dordrecht: Kluwer),
269.
Gingold, R.A. & Monaghan, J.J. 1977, Mon. Not. R. Astron.
Soc., 181, 375.
Hernquist, L. & Katz, N. 1989, Ap.J.Supp., 70, 419.
Houser, J.L. 1996, Ph.D. Dissertation, Drexel University.
Lucy, L.B. 1977, Astron. J., 82, 1013.
Monaghan, J.J. 1988, Comp. Phys. Comm., 48, 89.
Monaghan, J.J. 1992, Ann. Rev. Astron. Astrophys., 30,
543.
Benz, W. 1990, in Numerical Modeling of Stellar Pulsation: Problems and Prospects, ed. J.R. Buchler (Dordrecht: Kluwer), 269.
Gingold, R.A. & Monaghan, J.J. 1977, Mon. Not. R. Astron. Soc., 181, 375.
Hernquist, L. & Katz, N. 1989, Ap.J.Supp., 70, 419.
Houser, J.L. 1996, Ph.D. Dissertation, Drexel University.
Lucy, L.B. 1977, Astron. J., 82, 1013.
Monaghan, J.J. 1988, Comp. Phys. Comm., 48, 89.
Monaghan, J.J. 1992, Ann. Rev. Astron. Astrophys., 30, 543.