Well-balanced schemes for the Euler equations with gravitation

Transcription

1 Well-balanced schemes for the Euler equations with gravitation Roger Käppeli Joint work with S. Mishra

2 Outline Introduction Well-balanced scheme for HydroStatic Equilibrium (HSE) First order Second order Multi-D & further extensions Conclusions Roger Käppeli, Lyon

3 Outline Introduction Well-balanced scheme for HydroStatic Equilibrium (HSE) First order Second order Multi-D & further extensions Conclusions Roger Käppeli, Lyon

4 i) Introduction Stellar life cycle

5 i) Introduction Stellar life cycle

6 i) Introduction Stellar life cycle

7 i) Introduction Core-collapse supernova General idea: Explosion powered by gravitational binding energy of forming compact remnant: GRAVITY BOMB! Roger Käppeli, Lyon 2013 Mass of remnant Radius of remnant 7

8 i) Introduction Core-collapse supernova Roger Käppeli, Lyon

9 i) Introduction Radial profile The problem: (in our simulations) Ability to maintain near hydrostatic equilibrium for a long time! Roger Käppeli, Lyon

10 Outline Introduction Well-balanced scheme for HydroStatic Equilibrium (HSE) First order Second order Multi-D & further extensions Conclusions Roger Käppeli, Lyon

11 ii) WB scheme for HSE 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) Roger Käppeli, Lyon

12 ii) WB scheme for HSE Well-balanced scheme for HSE (2) Classical solution algorithm: (Local) Lax-Friedrichs HLL (C) Roe Lax (1954), Rusanov (1961) Harten, Lax and van Leer (1983), Toro et al. (1994) Roe (1981) Roger Käppeli, Lyon

13 ii) WB scheme for HSE Well-balanced scheme for HSE (3) Interested in hydrostatic EoS: 13

14 ii) WB scheme for HSE Well-balanced scheme for HSE (3)

15 ii) WB scheme for HSE Well-balanced scheme for HSE (3)

16 ii) WB scheme for HSE Well-balanced scheme for HSE (3)

17 ii) WB scheme for HSE Well-balanced scheme for HSE (3)

18 ii) WB scheme for HSE Well-balanced scheme for HSE (4) The problem: (in our simulations) Ability to maintain near hydrostatic equilibrium for a long time! Roger Käppeli, Lyon

19 ii) WB scheme for HSE Well-balanced scheme for HSE (5) 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!!! Roger Käppeli, Lyon

20 ii) WB scheme for HSE Well-balanced scheme for HSE (5) 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 Roger Käppeli, Lyon 2013 scheme 20

21 ii) WB scheme for HSE Well-balanced scheme for HSE (6) Hydrostatic equilibrium Describes only a mechanical equilibrium... Density and pressure not uniquely determined Entropy Temperature Arbitrary entropy or temperature profiles not (physically) stable (convection!) Roger Käppeli, Lyon

22 ii) WB scheme for HSE Well-balanced scheme for HSE (7) Consider constant entropy profile Using the thermodynamic relation Enthalpy Hydrostatic eq.

23 ii) WB scheme for HSE Well-balanced scheme for HSE (8) Perform equilibrium reconstruction:

24 ii) WB scheme for HSE Well-balanced scheme for HSE (9) Well-balanced discretization of momentum source term Roger Käppeli, Lyon 2013 Well-balanced wrt hydrostatic equilibrium! 24

25 ii) WB scheme for HSE Well-balanced scheme for HSE (10) Second order extension: pressure, density Eq. perturbation Stencil: Data Equilibrium Similar to Botta et al. 2004,Fuchs et al Time stepping:

26 ii) WB scheme for HSE Example 1 Hydrostatic atmosphere in a constant gravitational field

27 ii) WB scheme for HSE Example 2 Hydrostatic atmosphere in a constant gravitational field

28 ii) WB scheme for HSE Example 3 Hydrostatic atmosphere in a constant gravitational field

29 Outline Introduction Well-balanced scheme for HydroStatic Equilibrium (HSE) First order Second order Multi-D & further extensions Conclusions Roger Käppeli, Lyon

30 ii) Multi-D & further extensions Multi-dimensional extension Straight forward directional application of HydroStatic Reconstruction Hydrostatic equilibrium: Roger Käppeli, Lyon

31 ii) Multi-D & further extensions 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:

32 ii) Multi-D & further extensions Example 4

33 ii) Multi-D & further extensions Example 4

34 ii) Multi-D & further extensions Example 4

35 ii) Multi-D & further extensions Example 4

36 ii) Multi-D & further extensions Example 4

37 ii) Multi-D & further extensions Example 4

38 ii) Multi-D & further extensions Example 4

39 ii) Multi-D & further extensions Well-balanced scheme for HSE Consider constant temperature profile Using the thermodynamic relation Hydrostatic eq. Gibbs free energy

40 ii) Multi-D & further extensions Well-balanced scheme for HSE Interested in numerical hydrostatic equilibrium: Roger Käppeli, HYP2012, Padova 40

41 ii) Multi-D & further extensions Well-balanced scheme for HSE

42 ii) Multi-D & further extensions Well-balanced scheme for HSE

43 ii) Multi-D & further extensions Well-balanced scheme for HSE

44 ii) Multi-D & further extensions Example 5 Hydrostatic atmosphere in a constant gravitational field

45 Conclusions 1D well-balanced scheme for (isentropic, isothermal, arbitrary) hydrostatic equilibrium (for general EoS) Extension to higher-order? Non-zero velocity steady state? (e.g. for steady accretion flow) Multi-D well-balanced scheme for (isentropic, isothermal) hydrostatic equilibrium (for general EoS) Käppeli, Lyon 2013 Thank youroger for your attention!!! 45