Well-balanced schemes for the Euler equations with gravitation

Transcription

Well-balanced schemes for the Euler
equations with gravitation
Roger Käppeli
Joint work with S. Mishra
Outline
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Introduction
Well-balanced scheme for HydroStatic
Equilibrium (HSE)
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First order
Second order
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Multi-D & further extensions
●
Conclusions
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Roger Käppeli, Lyon 2013
2
Outline
●
●
Introduction
Equilibrium (HSE)
●
●
First order
Second order
●
●
Conclusions
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3
i) Introduction
Stellar life cycle
i) Introduction
Stellar life cycle
i) Introduction
Stellar life cycle
i) Introduction
Core-collapse supernova
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General idea:
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Explosion powered by gravitational binding energy
of forming compact remnant:
GRAVITY BOMB!
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Mass of remnant
Radius of remnant
7
i) Introduction
Core-collapse supernova
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i) Introduction
Radial profile
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The problem: (in our simulations)
Ability to maintain near hydrostatic equilibrium for a long time!
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Outline
●
●
Introduction
Equilibrium (HSE)
●
●
First order
Second order
●
●
Conclusions
08.10.13
10
ii) WB scheme for HSE
Well-balanced scheme for HSE
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Consider 1D hydrodynamics eqs with gravity
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Classical solution algorithm:
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Solve homogeneous eqs with Godunov type method
(i.e. solve Riemann problem)
Account for source term in second step (split/unsplit)
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Well-balanced scheme for HSE (2)
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Classical solution algorithm:
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(Local) Lax-Friedrichs
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HLL (C)
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Roe
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Lax (1954), Rusanov (1961)
Harten, Lax and van Leer (1983), Toro et al. (1994)
Roe (1981)
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Interested in hydrostatic
EoS:
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●
The problem: (in our simulations)
Ability to maintain near hydrostatic equilibrium for a long time!
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Solutions:
at each time
–
–
Steady state preserving reconstructions, wellbalanced schemes
e.g. LeVeque (1998), LeVeque & Bale (1998), Botta et al. (2004),
Fuchs et al. (2010), Xing & Shu (2013), Vides et al. (2013)
Note: there are many, many more... especially for shallow-water eqs!!!
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●
Solutions:
at each time
–
–
Steady state preserving reconstructions, wellbalanced schemes
e.g. LeVeque (1998), LeVeque & Bale (1998), Botta et al. (2004),
Fuchs et al. (2010), Xing & Shu (2013), Vides et al. (2013)
Requirements
Equilibrium not known in advance (self-gravity)
● Extensible for general EoS
● (At least) second order accuracy
●Preserve
Note:
there are robustness
many, many more...
especially
shallow-water eqs!!!
of base
shock for
capturing
08.10.13
scheme
●
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Hydrostatic equilibrium
Describes only a mechanical equilibrium...
Density and pressure not uniquely determined
Entropy
Temperature
Arbitrary entropy or temperature profiles not
(physically) stable (convection!)
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Consider constant entropy profile
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Using the thermodynamic relation
Enthalpy
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Hydrostatic eq.
Perform equilibrium reconstruction:
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Well-balanced discretization of momentum source term
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08.10.13
Roger Käppeli,
Lyon 2013
Well-balanced wrt
hydrostatic
equilibrium!
24
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Second order extension:
pressure, density
Eq. perturbation
Stencil:
Data
Equilibrium
Similar to Botta et al. 2004,Fuchs et al. 2010
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Time stepping:
Example 1
Hydrostatic atmosphere in a constant gravitational field
Example 2
Example 3
Outline
●
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Introduction
Equilibrium (HSE)
●
●
First order
Second order
●
●
Conclusions
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ii) Multi-D & further extensions
Multi-dimensional extension
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Straight forward directional application of
HydroStatic Reconstruction
Hydrostatic equilibrium:
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Example 4
Polytrope: model star (e.g. main sequence stars, white dwarfs,
neutron stars)
HSE:
Poisson equation:
Equation of state
Take
~ neutron stars
Then there's an exact solution:
Example 4
Example 4
Example 4
Example 4
Example 4
Example 4
Example 4
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Consider constant temperature profile
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Using the thermodynamic relation
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Hydrostatic eq.
Gibbs free energy
Interested in numerical hydrostatic
equilibrium:
28.06.12
Roger Käppeli, HYP2012, Padova
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Example 5
Conclusions
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1D well-balanced scheme for (isentropic, isothermal,
arbitrary) hydrostatic equilibrium (for general EoS)
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Extension to higher-order?
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Non-zero velocity steady state?
(e.g. for steady accretion flow)
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Multi-D well-balanced scheme for (isentropic,
isothermal) hydrostatic equilibrium (for general EoS)
08.10.13
Käppeli, Lyon 2013
Thank youRoger
for
your attention!!!
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Well-balanced schemes for the Euler equations with gravitation

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