SMOOTHED PARTICLE HYDRODYNAMICS
J.J. MONAGHAN
Department of Mathematics, Monash University,
Clayton, Victoria 3168, Australia
In this review of SPH we describe advances made in the last
few years. SPH is now seen as one of the simplest of the meshless numerical
methods. Despite its simplicity it is a proven, robust tool for investigating
uid dynamics with or without gravity. One of the desirable features of SPH
is that the resolution can vary with both space and time. This has now
been extended to include variation with direction. The dissipation terms
have been improved and their relation to Riemann solutions claried. For
relativistic blast waves and similar problems SPH gives results comparable
to the best Riemann and FCT methods. New applications include impact
and fragmentation, dusty gases, and nearly incompressible uids. A wide
range of problems in planetesimal formation and collision, engineering and
geophysics can now be studied using SPH.
Abstract.
1.
Introduction
SPH is a Lagrangian particle method which has now been used for many
problems in astrophysics. Its main attraction is its simplicity, the ease with
which complicated physics can be introduced, and the fact that it is very
forgiving.
In this review I will report on some of the improvements in the formulation of SPH, on new applications and make some comments about the
use of new hardware such as GRAPE and BEOWULF congurations. The
improvements in formulation come from simulations of relativistic gas dynamics (Chow and Monaghan 1997) where Riemann solvers have led the
way, and from work on the automatic adjustment of the resolution (Shapiro
et al. 1996). The new applications range from the ground breaking work
of Benz and Asphaug (1994) on fracture and fragmentation, and to dusty
gas dynamics (Monaghan 1997a). There has also been a great deal of in-
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J.J. MONAGHAN
teresting work on problems in engineering and geology which, while not
directly applicable to astrophysics, may suggest solutions to astrophysical
problems.
I have noticed that many of the people that use SPH are often intimidated by the wonderful results achieved by nely tuned FCT and Riemann
solvers applied to simple gas dynamics. I thought I might start this review
by commenting on some of the problems with these nite dierence methods. It is already clear that SPH has an obvious advantage in the ease with
which resolution in space and time can be introduced. A further consideration is that some Riemann methods can fail. For example linear Riemann
solvers fail where large regions of low density are suddenly created, as in the
Sjorgreen test (Einfeldt et al. 1991). When a shock moves slowly through
the nite dierence mesh (Roberts 1990) signicant down stream noise can
be generated. Colella and Woodward (1984) spend one third of their paper
discussing an articial viscosity to get rid of this problem. Exact, or good,
non-linear Riemann solvers do not have these problems, but you then need
to solve the Riemann problem and, in physically complicated problems,
this is a substantial undertaking. No Riemann solutions have yet been obtained for the important class of problems where chemical reactions occur
and more than one phase exists. Other defects of Riemann methods are
discussed by Quirk (1994) in a fascinating paper.
2.
2.1.
Advances in Formulation
DISSIPATION, SHOCKS AND RELATIVITY
The standard form of viscous dissipation in SPH was constructed by analogy with the dissipation in gases and then a quadratic term was included to
prevent interpenetration of gas streams colliding with high relative velocity.
While this works satisfactorily it gives no clue as to how a dissipation term
should be constructed for relativistic gas dynamics. The solution of this
problem was indirectly due to the Riemann solver nite dierence solution
simulation of relativistic gas dynamics by Marti and Muller (1996).
It is simpler to talk about the non-relativistic case. The Riemann solver
formulation involves a dissipative term associated with eigenvalues and
jumps in the momentum, energy and density across a cell boundary. The
natural way to mimic these terms is to replace the eigenvalue by a signal
velocity vsig and the jumps in a variable by the dierence of the variable
between the two SPH particles being considered (Monaghan 1997a). Rather
than use the thermal energy equation an equation for the energy/unit mass
(kinetic and thermal energy) was used. In non-relativistic calculations there
is an advantage in using the thermal energy equation because if the kinetic
energy dominates small errors in the velocity can have severe eects on
SMOOTHED PARTICLE HYDRODYNAMICS
359
Figure 1.
Velocity against distance for the non-relativistic blast wave using the new
dissipation and the momentum and energy equations. Initial particle spacing is 0.001.
the thermal energy. However, if you wish to simulate relativistic gases, the
natural variables are the momentum and energy.
Following the Riemann ideas the acceleration equation takes the form
X
dva
= 0 mb
dt
b
Pa Pb
+ 2
2a
b
0
!
Kvsig (a; b)
vab 1 j
ab
raWab ;
(2.1)
Where the non-standard SPH quantities are K a constant 1, vsig is a
signal velocity replacing the Riemann eigenvalues, and j is a unit vector
from b to a. The signal velocity is the velocity (measured in the computing
frame) with which sound waves from a and b approach each other. In the
simplest approximation this is
vsig (a; b) = ca + cb 0 vab 1 j:
(2.2)
If you now look at the viscous dissipation you will see that we have automatically included the quadratic term which was obtained previously by an
argument about stopping interpenetration using a kinetic pressure term.
By combining these equations with the continuity equation very good results have been achieved for the blast wave, Roberts and Sjogreen problems
(Monaghan 1997a). The only example we show here is the one dimensional
blast wave problems with zero initial velocity, initial pressure to the left of
the interface 1000 and on the right 0.01, density 1.0 and = 1:4. Typical
results are shown in Figures 1 and 2.
While the results for the non-relativistic problem are similar to those
with standard SPH dissipation terms the new formulation comes into its
own for the relativistic case. We will not give the details but conne ourselves to one example, the relativistic blast wave. We assume the gas is
composed of one type of baryon with rest mass m0 , the energy per unit
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J.J. MONAGHAN
Figure 2.
Figure 3.
Density against distance for the non-relativistic blast wave.
Baryon number density for the relativistic blast wave. Particle spacing 0:001.
mass is scaled with c2 , and the velocity with c. The initial conditions are
P = 1000 to the left and P = 0:01 to the right with baryon number density N = 1 everywhere. These are the same scaled values as for the nonrelativistic problem. The simulation uses a resolution varying in space. The
results for the baryon number density is shown in Figure 3. These results
are comparable in accuracy those obtained with the best Riemann solvers,
LCPFCT, and RHHLE and substantially better than those obtained using
SHASTA.
2.2.
ADAPTIVE RESOLUTION
SPH with h varying in space and time has been used for some time. However, it would clearly be desirable to have a dierent resolution in some
directions than in others e.g. when there is a shock along one of the axes.
Signicant advances in achieving this goal have been made by Shapiro et
SMOOTHED PARTICLE HYDRODYNAMICS
361
Figure 4. ASPH calculation of a one dimensional Zeldovich Pancake on the left, and a
standard SPH calculation on the right.
al. (1996) and Owen et al. (1998) and incorporated in an ASPH code. The
basic idea is to work with ellipsoidal kernels where the principal axes of
the ellipsoid are orientated according to the motion of neighbouring particles. In the case of the spherically symmetric kernel, with single smoothing
length h, it is common to determine h from the equation
1 r 1 v
d 1
=
;
(2.3)
dt
h
h
d
where d is the number of dimensions where I have used an equation for 1=h
rather than for h. The r 1 v takes into account the average eect of the
relative motion of the particles. In ASPH a matrix G replaces h. G has
eigenvalues 1=hk where hk , with k = 1; 2; 3 are the smoothing lengths along
the principal axis of an ellipsoid.
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J.J. MONAGHAN
In test calculations involving shock tubes, blast waves, Zeldovich pancakes and rotating disks the ASPH code works extremely well. In these
problems it is much more ecient than a standard SPH code. In Figure
4 a Zeldovich-pancake simulation. The gure (and those that follow are
taken from Owen et al. 1998) shows the ellipsoidal contours of the ASPH
simulation adjusting to the compression along the vertical axis.
Because the gradient of an ellipsoidal kernel is not along the line joining
the particles the angular momentum is not conserved exactly. In problems
involving disks the loss of conservation of total angular momentum is small
(3 percent over the time for one rotation period of the outer layers and several rotation periods of the core). However if the material changes density
rapidly the velocity gradients are not calculated accurately and G has very
large errors.
3.
3.1.
New Applications
IMPACT AND FRAGMENTATION
When planetesimals collide they may fragment. The same physics occurs in
rotating ore mixers and grinders. It is straightforward to include strength
into the SPH formulation of a perfectly elastic solid and that has been done
by a number of researchers (see for example Libersky and Petschek 1991).
This only requires including terms which enable the rate of change of the
elastic stress to be calculated in terms of the strain tensor. The uid stress
tensor is given by
ij = 0P ij + sij ;
(3.4)
where P is the pressure and sij is the traceless deviatoric stress tensor. In
the Hookian approximation
dsij
1
= 2(_ij 0 ij _ ) + rotation terms
(3.5)
dt
3
where is the shear modulus and the strain tensor is dened by
_ij = @vi =@xj + @vj =@xi :
(3.6)
Benz and Asphaug (1994) begin with the elastic model and use a quantitative model of fracture damage due to Grady and Kipp (1980). The
formulation depends on a damage factor D which enters the denition of
the stress tensor in the form
ij = 0P 3 ij + (1 0 D )sij
(3.7)
where P 3 = P if P > 0 and P 3 = (1 0 D)P if P < 0. The damage factor
changes with time according to the local strain. Because the damage is a
lagrangian quantity it is ideally suited to an SPH simulation.
SMOOTHED PARTICLE HYDRODYNAMICS
363
Figure 5.
Density (vertical axis) at the SPH gas and dust particles for a layer of dust
falling through air plotted against vertical distance x (horizontal axis). Upper symbols
show the dust layer, lower symbols the gas atmosphere. In agreement with theory there
is a slight change in the slope of the air density due to the dust layer falling through it.
Further details in (Monaghan 1997b).
In their ground breaking paper Benz and Asphaug (1994) rst showed
that SPH gives good results for the ideal elastic material and for crack
propagation When applied to the fragmentation case, SPH gives the best
results ever achieved for these problems (see also Asphaug et al. 1998).
3.2.
CLUMPING AND THE TENSILE INSTABILITY
In impact problems involving solids, the pressure can become negative.
This is the physical eect of the forces between the atoms. When the solid
is pulled (tensile stress) the atoms move further apart and the elastic contribution to the pressure becomes negative. When this happens the standard
forms of the pressure gradient terms in SPH become negative and the particles tend to clump (Swegle et al. 1995). This was already seen in early MHD
calculations (Phillips and Monaghan 1985). The clumping produces errors
in the estimates of the fragments in an impact problem so it would be better
to remove the problem. It also means that particles forced by some means
to be within a distance r satisfying r=h < 1=3 in an ideal gas will tend to
clump. There has been no entirely satisfactory solution to this problem for
the fragmentation of solids (but see Libersky and Randles (1996) who use
dissipation and Thomas and Couchman (1992) who truncate the gradient
of the kernel so that for r=h < 1=3 the gradient of the kernel has the same
value as at r=h = 1=3.
The same problem in MHD simulations can be evaded by working with
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J.J. MONAGHAN
the J 2 B force although momentum is not then conserved exactly. It would
be clearly be useful to be able to solve this tensile instability for both the
MHD and the impact problems.
3.3.
DUSTY GAS
Once heavy elements have been made and dust can form we need to take
into account the eect of dust on the gas. In the early and middle stages
of star formation the eect of dust is primarily through the radiation attempting to drive the dust from the system. The drag between the dust
and the gas allows the force to be transferred to the gas.
In many astrophysical problems the drag is enormous so it is natural to
use implicit methods. Implicit methods have been tried with SPH for gas
ow but they are cumbersome and so inecient they are not worth using.
Because the drag involves pair interactions it turns out that by working with
each pair a very ecient implicit scheme can be worked out (Monaghan
1997b).
This technique has been tested for two basic problems (Monaghan
1997b, all in one dimension) wave propagation and dust layers falling under gravity through air. In Figure 6 we show the density of the dust and gas
particles in a one dimensional simulation of a layer of dust falling through
gas in hydrostatic equilibrium under gravity. The SPH simulation accurately predicts the slight change in gas density gradient within the moving
dust layer, and the motion of the dust layer.
3.4.
RADIATIVE TRANSFER AND CONDUCTION
Radiative transfer in the conduction form involves second derivative terms
which require some care in an SPH formulation. The appropriate formulation has been known for some time. What isn't so well known is how
to handle the case where there is a rapid jump in the eective thermal
conductivity. The answer is surprisingly simple.
When the thermal conductivities changes discontinuously the ux of
heat must still be continuous. By taking this into account the appropriate
change is to replace the sum of the thermal conductivities according to
a + b
by
4a b
a + b
(3.8)
Amazingly this gives excellent results even when the thermal conductivity
jumps by a factor 1000 from one particle to the next. An extensive series
of calculations have been made by Cleary and Monaghan (1998).
SMOOTHED PARTICLE HYDRODYNAMICS
3.5.
365
ACCRETION, STAR FORMATION AND DISKS
Standard SPH, combined perhaps with some of the improvements discussed
earlier, can be used to study the formation of stars. When a fragment in
a cloud begins to collapse the density and scale eventually requires such
small time steps that the calculation is paralysed. Bate et. al (1995) have
worked out a way to continue the calculation by replacing the dense region
by a single particle with appropriate boundary conditions. This allows the
evolution of the cloud to be followed as binary and other n-body systems
form.
SPH continues to be used as a tool to study the dynamics of disks.
The work of Artymowicz and Lubow (1994), Murray (1997) and Nelson
et al. (1998) are examples of the widespread application of SPH to these
problems.
4.
New Hardware
The GRAPE board was developed in Tokyo (Sugimoto et al. 1990) originally to calculate the gravitational forces on a set of particles by direct
summation. Its advantage is cheapness (a board plus a work station). The
summation was extended to work with a tree code (Makino and Funato
1993). They then extended to calculate SPH terms. Steinmetz (1996) has
given a detailed analysis of version of GRAPE for cosmological SPH simulations (GRAPESPH). He shows that (given the technology of 3 years
ago) it is possible to emulate results previously achieved by super computers. The situation is now more dicult to assess because the technology
is changing so rapidly. For example the introduction of cheap BEOWULF
congurations (http://qso.lanl.gov/msw).
5.
Summary
While there have been many applications of standard forms of SPH over
the last few years there have been generalizations which increase the efciency and extensions to deal with more complex problems. There has
been a big interest in SPH by computational engineers and computational
geophysicists because of the ease with which SPH can be applied to complicated geometry and complicated ow problems. Further developments can
be expected in these areas as well as in astrophysical problems associated
with relativistic ows, mixtures of solid particles and gas, radiative transfer
and MHD problems. As always, new computers, in this case probably the
BEOWULF congurations, will be a major resource.
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J.J. MONAGHAN
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