sph vs lagrange
TRANSCRIPT
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Lagrangian methods and SmoothedParticle Hydrodynamics (SPH)
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Eulerian Grid Methods
The methods covered so far inthis course use an Euleriangrid:
Prescribed coordinates
In `lab frame' Fluid elements flow
through grid zones
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Eulerian Grid Methods
This is probably the standardapproach to solving theequations of fluid motions inmost disciplines.
Many decades of research intosolution techniques
Extremely sophisticated, high-order accuracy methods
Can accurately describe verycomplex phenomena.
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Eulerian Grid Methods
However, simplestpossible dynamics faressurprisingly poorly
Even high-ordermethods are quitediffusive whenadvection over a largenumber of grid cells isnecessary
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Eulerian Grid Methods
This is fundamental to how
Eulerian grid codes work.
Can be ameliorated but not fixed.
Once some of a quantity enters agrid cell its contribution is spread
throughout domain through someaveraging procedure.
Higher order methods do this to alesser degree than lower-order
methods, but the effect remains.Occurs for any evolved quantity.
``Numerical diffusion''
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Advection-Dominated Flows
There are many systems in
astrophysics which aredominated by large-scaleadvection of fluid
Eulerian grid is not necessarilythe most natural approach in
these systems
Cosmology: evolution isdominated by large scale fallingof material onto local density
enhancements
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Advection-Dominated Flows
There are many systems in
astrophysics which aredominated by large-scaleadvection of fluid
Eulerian grid is not necessarilythe most natural approach in
these systems
Accretion disks: Flow isdominated by differentialKeplarian rotation around
central object
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
1D Lagrangian Formulation
Lagrangian formulation
given in lecture 4No fluxes `through' fluid
element interfaces, asno transport through
interfaces*
Typically implemented ona staggered grid:
No purely advective fluxes!
*
absent other physics like dissipative transport
vi-1
p,ei
vi+1
p,ei-1
vi
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
1D Lagrangian Formulation
Huge benefit: openboundaries mesh canexpand as necessary
And, of course, nonumerical diffusion frompurely moving fluidaround
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
1D Lagrangian Formulation
These advantages make1d lagrangian gridmethods a naturalchoice for applicationssuch as stellar evolution
Typically use `masscoordinates' evenoutside of numerical
context
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Multi-D Lagrangian Gridding
Works extremely well in
1d.
In multidimensions,more complexitypossible in geometry
Even differentialrotation / shear canlead to disasterously
tangled meshes.More complex motions
almost hopeless
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Multi-D Lagrangian GriddingCan deal with this
problem byremeshing every sooften
Remeshing can be a
very expensive step(choosing an optimalnew mesh for a set ofpoints is difficult)
Loss of main benefit ofLagrangian method diffusive (as fluid`moves through'remeshing)
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Grid is a way of assigningneighbors to structure localinteractions.
If can determine localneighbors without discretizing
on a grid can avoid the issueswith tangling/remapping
Astrophysics has a long (>60yr) history with one gridlessmethod gravitational N-bodycalculations
Gridless methods
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
It is clearly true that one can
write the density field in adomain as integral over infinitenumber of point particles:
N-body calculationapproximates this by using afinite number of particles
N-body calculation
http://www.physics.drexel.edu/~steve/n-body.html
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Smoothed Particle Hydrodynamics
For hydrodynamics, interactions need to be local
Quantities stored at N `free' particles
If infinite number of particles, any hydrodynamicquantity A could be defined as
Finite number of particles quantities must be
smoothed over some finite smoothing length h
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Properties of smoothing function W(r,h)
In small h limit, goes to delta function (usually) symmetric about r=0
Compact support W is exactly zero outside ofsome finite radius around the particle
Cubic spline is typical choice
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
SPH `Discretization' Error
Even in this limit, smoothed quantity has errorO(h2) [Why?]
Very difficult (likely impossible) to have robust,stable smoothing with higher order accuracy.
With finite number of particles,
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
SPH `Discretization' Error
Taylor expand this around particle `a's position,and define W
ab= W(r
a r
b,h)
Even for constant function (say, A = 1), not
guaranteed exact; must divide by first term, egdo SPH interpolation of A / SPH interpolation of 1.
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
SPH `Discretization' Error
Error comes from discretization finite number of particles
No guarantee that there will beenough particles well enoughdistributed so that
although corrections can be made
This problem is much worse forderivatives; numericalderivatives of `noisy' data known
hard problem.
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Lagrangian Formulation
Can now express other equations in terms ofLagrangian, Hamiltonian dynamics
Lagrangian for Hydrodynamics is
from Euler-Lagrange equations, get eqn of motion
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Momentum Equation
Note symmetry; contribution to momentum ofparticle a from b equal and opposite to b from a
Conserves momentum exactly
This form of gradient of pressure has somediscreteness inaccuracies, but the symmetry ismuch more important
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Energy Equation
Similarly, Hamiltonian can be written
implying a specific energy equation
again, note symmetry.
Because of troubles with internal energy evolutionin high-speed flows, some SPH practitionersevolve entropy rather than energy; applies to gridcodes too.
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Setting Initial Conditions in an SPH code
Unlike grid code (ICs are set everywhere in
domain), have to sample (typically randomly) thedensity profile and put particles there
If under-resolving the density profile is a problem,some caution is necessary; fluctuations causedby particle placement can be a problem. Mayhave to relax initial conditions.
Cosmology: extreme care needed --- any initial
fluctuations in density become large scalestructure! Specialized techniques.
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Time Evolution
i li i l i
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Visualization/Analysis
Grid code: output is a complete description of
hydrodynamic states throughout domain.
Easy to visualize, analyze many instantaneousquantities much harder to examine histories offluid elements.
SPH: Opposite problem; can see what happens toany one fluid parcel immediately, but need someanalysis tools even just to make a picture of what
the domain looks like
Vi li i /A l i
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Visualization/Analysis
Create a grid, use SPH interpolation tofind quantities at each point on grid
Complications
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
Complications(Things I really should talk about but don't
have time) Need artificial viscosity to handle shocks
Real world necessities variable timesteps andsmoothing lengths complicate or break some of
SPHs nice qualities There are a variety of techniques for finding
neighbors within some distance h ; one commontechnique, using tree searches, integrates very
nicely with using the same tree for gravitationalN-body solving, making SPH + treecode gravity avery natural match.
SPH G id C d
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
SPH vs Grid Codes Handes open, free
boundaries muchbetter
Much less diffusion forbulk motion
Automatically resolveshigh density region
No need to wastecomputation on empty
space
Couples naturally to N-body gravity
Very robust
Poor at dealing with
shocks Low-order spatial
accuracy
Derivatives harder,
making some physics(MHD) harder
More caution requiredwith initial conditions
Hard to followinteresting dynamicsin low-density regions
Too robust?
SPH
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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi
SPH resources
References:
Price (2005) astro-ph/0507472: Thesis chapter,very good review
Codes
GADGET-2: Robust, widely-used SPH code http://www.mpa-garching.mpg.de/gadget/
StarCrash:
http://www.astro.northwestern.edu/StarCrash/ SuperSPHplot: visualization tool
http://www.astro.ex.ac.uk/people/dprice/supersphplot/
http://www.mpa-garching.mpg.de/gadget/http://www.astro.ex.ac.uk/people/dprice/supersphplot/http://www.astro.ex.ac.uk/people/dprice/supersphplot/http://www.mpa-garching.mpg.de/gadget/