sph vs lagrange

8/7/2019 sph vs lagrange

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Computation in Astrophysics Seminar(Spring 2006)L. J. Dursi

Lagrangian methods and SmoothedParticle Hydrodynamics (SPH)


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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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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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However, simplestpossible dynamics faressurprisingly poorly

Even high-ordermethods are quitediffusive whenadvection over a largenumber of grid cells isnecessary


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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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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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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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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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Huge benefit: openboundaries mesh canexpand as necessary

And, of course, nonumerical diffusion frompurely moving fluidaround


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These advantages make1d lagrangian gridmethods a naturalchoice for applicationssuch as stellar evolution

Typically use `masscoordinates' evenoutside of numerical

context


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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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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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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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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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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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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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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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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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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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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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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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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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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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Time Evolution

i li i l i


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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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Visualization/Analysis

Create a grid, use SPH interpolation tofind quantities at each point on grid

Complications


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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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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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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/

sph vs lagrange

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