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A Well-Balanced Multi-Dimensional Reconstruction Scheme for - - PowerPoint PPT Presentation

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A Well-Balanced Multi-Dimensional Reconstruction Scheme for Hydrostatic Equilibria Roger Kppeli Outline Introduction (Astro)Physical motivation Well-balanced scheme for HydroStatic Equilibrium (HSE) First order Second

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A Well-Balanced Multi-Dimensional Reconstruction Scheme for Hydrostatic Equilibria

Roger Käppeli

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28.06.12 Roger Käppeli, HYP2012, Padova 2

Outline

  • Introduction
  • (Astro)Physical motivation
  • Well-balanced scheme for HydroStatic

Equilibrium (HSE)

  • First order
  • Second order
  • Multi-dimensional extension
  • Limitations
  • Conclusion
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i) Introduction

Stellar life cycle

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Stellar life cycle

i) Introduction

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Stellar life cycle

i) Introduction

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28.06.12 Roger Käppeli, HYP2012, Padova 6

Core-collapse supernova

  • General idea:
  • Explosion powered by gravitational binding energy
  • f forming compact remnant:

Mass of remnant Radius of remnant

GRAVITY BOMB!

i) Introduction

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28.06.12 Roger Käppeli, HYP2012, Padova 7

Core-collapse supernova

i) Introduction

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28.06.12 Roger Käppeli, HYP2012, Padova 8

Radial profile

  • The problem: (in our simulations)

Ability to maintain near hydrostatic equilibrium for a long time!

i) Introduction

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28.06.12 Roger Käppeli, HYP2012, Padova 9

Outline

  • Introduction
  • (Astro)Physical motivation
  • Well-balanced scheme for HydroStatic

Equilibrium (HSE)

  • First order
  • Second order
  • Multi-dimensional extension
  • Limitations
  • Conclusion
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28.06.12 Roger Käppeli, HYP2012, Padova 10

Well-balanced scheme for HSE

  • Consider 1D hydrodynamics eqs with gravity
  • Classical solution algorithm:
  • Solve homogeneous eqs with Godunov type method

(i.e. solve Riemann problem)

  • Account for source term in second step (split/unsplit)

ii) WB scheme for HSE

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28.06.12 Roger Käppeli, HYP2012, Padova 11

Well-balanced scheme for HSE (2)

  • Classical solution algorithm:
  • (Local) Lax-Friedrichs
  • HLL (C)
  • Roe

Harten, Lax and van Leer (1983), Toro et al. (1994) Roe (1981) Lax (1954), Rusanov (1961)

ii) WB scheme for HSE

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Well-balanced scheme for HSE (3)

Interested in hydrostatic EoS:

ii) WB scheme for HSE

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Well-balanced scheme for HSE (3)

ii) WB scheme for HSE

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Well-balanced scheme for HSE (3)

ii) WB scheme for HSE

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Well-balanced scheme for HSE (3)

ii) WB scheme for HSE

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Well-balanced scheme for HSE (3)

ii) WB scheme for HSE

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28.06.12 Roger Käppeli, HYP2012, Padova 17

Well-balanced scheme for HSE (4)

  • The problem: (in our simulations)

Ability to maintain near hydrostatic equilibrium for a long time!

ii) Numerical models & methods

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28.06.12 Roger Käppeli, HYP2012, Padova 18

Well-balanced scheme for HSE (5)

  • Solutions:

–

at each time

– Steady state preserving reconstructions, well-

balanced schemes

ii) Numerical models & methods

Note: there are many, many more... especially for shallow-water eqs!!!

e.g. LeVeque (1998), LeVeque & Bale (1998), Botta et al. (2004), Fuchs et al. (2010)

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28.06.12 Roger Käppeli, HYP2012, Padova 19

Well-balanced scheme for HSE (5)

  • Solutions:

–

at each time

– Steady state preserving reconstructions, well-

balanced schemes

ii) Numerical models & methods

e.g. LeVeque (1998), LeVeque & Bale (1998), Botta et al. (2004), Fuchs et al. (2010)

Note: there are many, many more... especially for shallow-water eqs!!!

Requirements

  • Equilibrium not known in advance (self-gravity)
  • Extensible for general EoS
  • (At least) second order accuracy
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28.06.12 Roger Käppeli, HYP2012, Padova 20

Well-balanced scheme for HSE (6)

Interested in numerical hydrostatic equilibrium:

ii) Numerical models & methods

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Well-balanced scheme for HSE (6)

ii) Numerical models & methods

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Well-balanced scheme for HSE (6)

ii) Numerical models & methods

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Well-balanced scheme for HSE (6)

ii) Numerical models & methods

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Well-balanced scheme for HSE (7)

  • Second order extension:
  • Time stepping:

Reconstruction in deviation from equilibrium

Similar to Botta et al. 2004,Fuchs et al. 2010

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Example 1

Hydrostatic atmosphere in a constant gravitational field

ii) WB scheme for HSE

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Example 2

Hydrostatic atmosphere in a constant gravitational field

ii) WB scheme for HSE

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Example 3

Hydrostatic atmosphere in a constant gravitational field + small amplitude waves

ii) WB scheme for HSE

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Example 3 (2)

Hydrostatic atmosphere in a constant gravitational field + large amplitude waves

ii) WB scheme for HSE

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28.06.12 Roger Käppeli, HYP2012, Padova 29

Example 6

Polytrope: model star Euler equations in spherical symmetry: Poisson equation in spherical symmetry:

(e.g. main sequence stars, white dwarfs, neutron stars)

ii) WB scheme for HSE

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Example 6 (2)

Polytrope: model star

HSE: Poisson: ~ neutron stars

ii) WB scheme for HSE

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Example 6 (3)

Polytrope: model star

~ neutron stars

+ density perturbation

ii) WB scheme for HSE

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28.06.12 Roger Käppeli, HYP2012, Padova 32

Outline

  • Introduction
  • (Astro)Physical motivation
  • Well-balanced scheme for HydroStatic

Equilibrium (HSE)

  • First order
  • Second order
  • Multi-dimensional extension
  • Limitations
  • Conclusion
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28.06.12 Roger Käppeli, HYP2012, Padova 33

Multi-dimensional extension

  • Straight forward directional application of

HydroStatic Reconstruction

  • Numerical equilibrium:

3D analogous...

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Example 7

Polytrope: model star

(e.g. main sequence stars, white dwarfs, neutron stars)

~ neutron stars Take Then there's an exact solution: HSE: Poisson equation: Equation of state

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Example 7

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Example 7

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28.06.12 Roger Käppeli, HYP2012, Padova 37

Conclusions

  • 1D well-balanced scheme for hydrostatic

equilibrium (for any Equation of State EoS)

  • Extension to higher-order? Non-zero velocity

steady state?

  • Multi-D well-balanced scheme for hydrostatic

equilibrium

  • Unfortunately with limitations (so far...)

Although not exactly well-balanced for general EoS, the ability to maintain HSE is greatly increased Thank you for you attention!!!