Numerical Cosmology & Galaxy Formation€¦ · Lecture 6: Hydro schemes - Grid codes. Outline of the lecture course ... • The solution is almost always sought using numerical methods

Numerical Cosmology& Galaxy Formation

1

Benjamin Moster

Lecture 6: Hydro schemes - Grid codes

Outline of the lecture course

• Lecture 1: Motivation & Historical Overview

• Lecture 2: Review of Cosmology

• Lecture 3: Generating initial conditions

• Lecture 4: Gravity algorithms

• Lecture 5: Time integration & parallelization

• Lecture 6: Hydro schemes - Grid codes

• Lecture 7: Hydro schemes - Particle codes

• Lecture 8: Radiative cooling, photo heating

• Lecture 9: Subresolution physics

• Lecture 10: Halo and subhalo finders

• Lecture 11: Semi-analytic models

• Lecture 12: Example simulations: cosmological box & mergers

• Lecture 13: Presentations of test simulations

2 Numerical Cosmology & Galaxy Formation 5 18.05.2016

Computational Cosmology

• Cosmological model + initial conditions + simulation code = galaxies

SDSSCMB

generation theinitial conditions

running the simulation

analyzingthe data


Why hydrodynamics?

• Everything we see is gas or made from gas

• Need to follow the hydrodynamics: To form galaxies and starsTo study the interstellar, intergalactic and intracluster medium


The Bullet cluster

5

collisionlessdark matter

collisionlessdark matter

collisional gas

Numerical Cosmology & Galaxy Formation 6 25.05.2016

Why hydrodynamics?

• Problems are so complex that analytical methods are inadequate.

• The solution is almost always sought using numerical methods

• No single numerical recipe for all hydrodynamic problems

• Each problem may involve different additional physics: e.g. radiation-hydrodynamics or magneto-hydrodynamics

• Two broad categories: Eulerian methods (fixed position) Lagrangian methods (fixed element)


Eulerian vs Lagrangian methods


The Euler equations


On the blackboard…(see scanned notes)

The Euler equations


• The equations of hydrodynamics can be written in terms of

conserved quantities

Advection on a grid


• Let’s consider a simpler problem first and assuming u = const.

• discretize space into cells for numerical treatment

• need to advect the mass such that total mass is conserved➡ calculate mass fluxes at cell interfaces➡ remove mass from the cell on one side of the interface and add it to the cell on the other side (this ensures mass conservation)

Advection on a grid


Advection on a grid


• next higher order: piecewise linear within cell

•

• average density at interface over time step Δt

• Flux:

Advection on a grid


Advection on a grid


Advection on a grid


• Higher order schemes, such as the piecewise linear Lax-Wendroff

scheme shown here, produce oscillations near discontinuities

• Piecewise linear elements can have overshoots

• Successful method to prevent overshoots is the use of slope limitersModify the slope if this is necessary to prevent overshoots

Characteristics


• With a suitable advection scheme we can solvewith u = constant

• However, the full Euler equations are coupled:

Characteristics


• Can we decouple these equations?First we rewrite them (for simplicity in 1D) by defining:

• We then get

Characteristics


• The Euler equations in terms of q1,q2,q3 are then given by with

• Can be written more compact using the Jacobian

Characteristics


• Consider a simple system with A(q) = A = constant We get:

• The equations can then be decoupled by finding the eigensystem of

the matrix A

• Decompose the state q in the eigenbasisand find the equations for the individual components

Characteristics


• Using this decomposition with

• And therefore

• This is a simple advection equation with characteristic velocity λm

• We can thus solve the coupled set of equations by expanding the state vector in the eigenbasisadvecting each qm with its characteristic velocity recomputing the new state from the updated

Characteristics


• Each of these modes propagating with a characteristic velocity is

called a characteristic

• For the full Euler equations A(q) ≠ constant➙ eigenvectors depend on q (and thus position) ➙ no global decomposition of the state vector possible

• Locally we find:

Characteristics


• Decompose the state vector locally at each cell interface and advect

the components with their local characteristic velocities

Riemann problems


• What about shocks and contact discontinuities?➙ eigensystems differ significantly on both sides

• Consider the full Riemann problem e.g. for uL = uR = 0

Riemann problems


• Solutions are self-similar, i.e. they depend only on

Godunov method


• Assume piecewise constant fluid state

• Exactly solve Riemann problem at each interface(i.e. solve algebraic equations iteratively)

• Choose small enough time step, such that solutions of neighboring interfaces do not overlap

• Calculate new average of conserved quantities at end of time step

• Easy as flux f(q(x=x0)) is constant for self-similar q((x−x0)/(t−t0))

Godunov method


• Assume piecewise constant fluid state

• Exactly solve Riemann problem at each interface(i.e. solve algebraic equations iteratively)

• Choose small enough time step, such that solutions of neighboring interfaces do not overlap

• Calculate new average of conserved quantities at end of time step

• Easy as flux f(q(x=x0)) is constant for self-similar q((x−x0)/(t−t0))

For linear problems: same as advecting the components in the eigenbasis

But accounts for shocks and contact discontinuities

Cons: Diffusive as constant fluxes correspond to donor-cell advection

The MUSCL-Hancock scheme


• Create a 2nd order Godunov method by

- using a higher-order reconstruction(piecewise linear, piecewise parabolic)

- computing the left and right q values at the interface

- advance these values in half a time step

- use these values in the Riemann solver as if the state is constant on each side of the interface

- • used in many codes, e.g.: Ramses, Arepo

Roe’s linearized Riemann solver


• Exact Riemann solvers can be slow (e.g. for magneto-hydrodynamics)

• Alternatively linearise the problem at each interface by settingA(q) → A(qave) with qave: average value between left and right state

• A solution can then be found by decomposing the left and right

states into the eigenbasis of A(qave)

• And advecting the components with the corresponding characteristic

velocity➡ yields a solution in smooth parts of the flow ➡ every reasonable average value for qave should work there

Roe’s linearized Riemann solver


• It is possible to choose the average qave such that the linearized

Riemann solver also gives the correct propagation of contact

discontinuities and shocks

• Roe average:

• When using this average one can show that the “jump” corresponds

exactly to one eigenvector with a eigenvalue given by the correct

velocity (e.g. the shock velocity).

Adaptive mesh refinement


• In AMR the local resolution is adapted according to refinement

criteria (usually high density)









Moving-mesh hydrodynamics


• Alternatively one can define a mesh based on a set of points using a Voronoi tessellation

• Points can be allowed to move, e.g. with the fluid ➙ then almost Lagrangian

• Procedure: use unspilt scheme transform to frame of moving interfacesolve Riemann problemtransform back









Pros: accurate hydro, automatic automatic refinement on accurate hydro refinement on density, density, Galilean invar., Galilean invariant conserves angular mom.

Cons:needs AMR to overhead (~30%) for Often problems with get to high mesh construction contact discontinuities resolution slow convergence

Up next

• Lecture 1: Motivation & Historical Overview

• Lecture 2: Review of Cosmology

• Lecture 3: Generating initial conditions

• Lecture 4: Gravity algorithms

• Lecture 5: Time integration & parallelization

• Lecture 6: Hydro schemes - Grid codes

• Lecture 7: Hydro schemes - Particle codes

• Lecture 8: Radiative cooling, photo heating

• Lecture 9: Subresolution physics

• Lecture 10: Halo and subhalo finders

• Lecture 11: Semi-analytic models

• Lecture 12: Example simulations: cosmological box & mergers

• Lecture 13: Presentations of test simulations


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Numerical Cosmology & Galaxy Formation€¦ · Lecture 6: Hydro schemes - Grid codes. Outline of the lecture course ... • The solution is almost always sought using numerical methods