Numerical Hydrodynamics: A Primer. Wilhelm Kley Institut für Astronomie & Astrophysik & Kepler Center for Astro and Particle Physics Tübingen

Transcription

1 A Primer Wilhelm Kley Institut für Astronomie & Astrophysik & Kepler Center for Astro and Particle Physics Tübingen Bad Honnef 2016

2 Hydrodynamics: Hydrodynamic Equations The Euler-Equations in conservative form read ρ + (ρ u) = 0 (1) (ρ u) + (ρ u u) = p + ρ k (2) (ρɛ) + (ρɛ u) = p u (3) u: Velocity, k: external forces, ɛ specific internal energy The equations describe the conservation of mass, momentum and energy. For completion we need an equation of state (eos): p = (γ 1) ρɛ (4) Using this and eq. (3), we can rewrite the energy equation as an equation for the pressure p + (p u) = (γ 1)p u (5) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

3 Hydrodynamics: Reformulating Expanding the divergences on the left side and use for the momentum and energy equation the continuity equation ρ + ( u )ρ = ρ u (6) u + ( u ) u = 1 ρ p + k (7) p + ( u )p = γp u (8) Since all quantities depend on space ( r) and time (t), for example ρ( r, t), we can use for the left side the total time derivative (Lagrange-Formulation). For example, for the density one obtains Dρ Dt = ρ + ( u )ρ = ρ u. (9) The Operator D Dt = + u (10) is called material derivative (equivalent to the total time derivative, d/dt). W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

4 Hydrodynamics: Lagrange-Formulation Use now the material derivative Dρ Dt D u Dt Dp Dt = ρ u (11) = 1 ρ p + k (12) = γp u (13) These equations describe the change of the quantities in the comoving frame = Lagrange-Formulation. For the Euler-Formulation, one analysed the changes at a specific, fixed point in space! The Lagrange-Formulation can be used conveniently for 1D-problems, for example the radial stellar oscillations, using comoving mass-shells. For the Euler-Formulation a fixed grid is used. W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

5 The problem Consider the evolution of the full time-dependent hydrodynamic equations. The non-linear partial differential equations of hydrodynamics will be solved numerically Continuum Discretisation W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

6 Method of solution Grid-based methods (Euler) Particle methods (Lagrange) fixed Grid - matter flows through grid ( ) u ρ + u u = p Methods: finite differences non-conservative Control Volume conservative Riemann-solver wave properties Problem: Discontinuities moving Grid/Particle - flow moved grid ρ d u = p dt Well known method: Smoothed Particle Hydrodynamics, SPH smeared out particles good for free boundaries, self-gravity W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

7 consider: 1D Euler equations describe conservation of mass, momentum and energy ρ + ρu x ρu + ρuu x ρɛ + ρɛu x ρ: density u: velocity p: pressure ɛ: internal specific energy (Energy/Mass) with the equation of state γ: adiabatic exponent partial differential equation in space and time need discretisation in space and time. = 0 (14) = p x = p u x (15) (16) p = (γ 1)ρɛ (17) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

8 Discretisation ψ Consider funktion: ψ(x, t) discretisation in space cover with a grid x = x max x min N ψ n j j 1 j j+1 cell average of ψ(x, t) at the grdipoint x j at time t n X ψj n = ψ(x j, t n ) 1 (j+1/2) x ψ(x, n t)dx x (j 1/2) x ψ n j is piecewise constant. j spatial index, n time step. W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

9 Integration in time Consider general equation ψ = L(ψ(x, t)) (18) with a (spatial) differential operator L. typical Discretisation (1. order in time), at time: t = t n = n t ψ ψ(t + t) ψ(t) t = ψn+1 ψ n t now at a special location, the grid point x j (with moving terms) = L(ψ n ) (19) ψ n+1 j = ψ n j + tl(ψ n k ) (20) L(ψk n ): discretised differential operator L (here explicit) - k in L(ψ k ): set of spatial indices: - typical for 2. order: k {j 2, j 1, j, j + 1, j + 2} (need information from left and right, 5 point Stencil ) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

10 Operator-Splitting A L i ( A), i = 1, 2 individul (Differential-)operators applied to the quantities A = (ρ, u, ɛ). Here, for 1D ideal hydrodynamics = L 1 ( A) + L 2 ( A) (21) L 1 : Advection L 2 : pressure, or external forces To solve the full equation the solution is split in several substeps A 1 = A n + tl 1 ( A n ) A n+1 = A 2 = A 1 + tl 2 ( A 1 ) (22) L i is the differential operator to L i. W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

11 advection-step In explicit conservation form ρ (ρu) (ρɛ) = ρu x = (ρuu) x = (ρɛu) x for u = (u 1, u 2, u 3 ) and f = (f 1, f 2, f 3 ) we have: u = (ρ, ρu, ρɛ) and f = (ρ u, ρuu, ρɛu). u + f ( u) = 0 (23) x This step yields: ρ n ρ 1 = ρ n+1, u n u 1, ɛ n ɛ 1 W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

12 Force terms Momentum equation u n+1 j = u j t u 1 ρ n+1 j = 1 p ρ x ( ) pj p j 1 x (24) for j = 2, N (25) energy equation ɛ = p u ρ x ( ) ɛ n+1 p j uj+1 u j j = ɛ j t ρ n+1 x j (26) for j = 1, N (27) on the right hand side we use the actual values for u, ɛ and p, i.e. here u 1, p 1, ɛ 1. This step yields: u 1 u n+1, ɛ 1 ɛ n+1 W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

13 Model equation for advection The continuity equation was Here, F m = ρu is the mass flow ρ + ρu x = 0 (28) Using the notation ρ ψ and u a = const. we obtain the Linear Advection Equation ψ + a ψ x = 0. (29) with a constant velocity a, the solution is a wave traveling to the right using ψ(x, t = 0) = f (x) we get ψ(x, t) = f (x at) Here f (x) is the initial condition at time t = 0, that is shifted by the advection with a constant velocity a to the right. The numerics should maintain this property as accurately as possible. W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

14 Linear Advection FTCS: Forward Time Centered Space algorithm ψ + a ψ x = 0 (30) ψ j 1 ψ j ψ j+1 Specify the grid : and write it follows ψ j n ψ x j n x = ψn+1 j j 1 t ψ n j = ψn j+1 ψn j 1 2 x x x j j+1 ψ n+1 j = ψj n a t ( ψ n 2 x j+1 ψj 1) n (31) (32) (33) The method looks well motivated: but it is unstable for all t! W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

15 Upwind-Method I or ψ ψ + aψ x = 0 (34) + a ψ x = 0 (35) ψ α a t a: constant (velocity) > 0 ψ(x, t) arbitrary transport quantity change of ψ in grid cell j ψ n+1 j x = ψ n j x + t(f in F out ) (36) in out The flux F in is for constant ψ j F in = a ψ n j 1 (37) F out = a ψ n j (38) Upwind-Method Information comes from upstream j 1 j j+1 purple regions will be transported into the next neighbour cell X W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

16 Upwind-Method II Extension for non-constant states ( F in = a ψ I x j 1/2 a t ) 2 (39) ψ α a t ψ I(x) ψ I (x) interpolation polynomial Here linear interpolation (straight line) This yields F in = a [ ψj 1 n + 1 ] }{{} 2 (1 σ) ψ j 1 (40) 1st Order } {{ } 2nd Order with σ = a t/ x ψ j ψ x x xj ψ I (x) = ψ n j in out j 1 j j+1 + x x j x ψ j (41) ψ j undivided differences 2nd order upwind ψ I (x) is evaluated in the middle of the purple area. X W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

17 Undivided Difference a) ψ j = 0 Upwind, 1st Order, piece-wise constant b) ψ j = 1 ( ) 2 ψj+1 ψ j 1 Fromm, centered derivative c) ψ j = ψ j ψ j 1 Beam-Warming, upwind slope d) ψ j = ψ j+1 ψ j Lax-Wendroff, downwind slope Often used is the 2nd Order Upwind (van Leer) Geometric Mean (maintains the Monotonicity) 2 (ψ j+1 ψ j )(ψ j ψ j 1 ) (ψ j+1 ψ j 1 ) if (ψ j+1 ψ j )(ψ j ψ j 1 ) > 0 ψ j = 0 otherwise (42) The derivatives are evaluated at the corresponding time step level or the intermediate time step W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

18 t t n+1 n+1/2 t n ψ n j 1 Using two steps: n+1 ψj ~ n+1/2 ψ ψ~ n+1/2 j 1/2 j+1/2 ψ n j Lax-Wendroff Method ψ n j+1 predictor-step (at intermediate time t n+1/2 ) Schematic overview of the method uses centered spatial and temporal differences that makes it 2nd order in space and time ψ n+1/2 j+1/2 = 1 ( ) ψj n + ψj+1 n σ ( ) ψj+1 n 2 2 ψn j The corrector-step (to new time t n+1 ) ψ n+1 j = ψ n j σ ( ψ n+1/2 j+1/2 ψ n+1/2 j 1/2 ) (43) (44) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

19 ψ(x) Example: Linar Advection Upwind - (Diffusive) 1 Linear Advection (N=600,t=40): Upwind, sigma=0.8 Analytic Upwind 0.8 Square function: Width 0.6 in the intervall [ 1, 1] velocity a = 1, until t = 40 periodic boundaries σ = a t/ x = (Courant no.) Van Leer Linear Advection (N=600,t=40): Van Leer, sigma=0.8 Analytic Van Leer Phi-Achse X-Achse Lax-Wendroff - (Dispersive) Linear Advection (N=600,t=40): Lax Wendroff, sigma=0.8 Analytic Lax Wendroff Phi-Achse 0.4 Phi-Achse X-Achse X-Achse W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

20 Stability analysis I Substitute for the solution a Fourier series (von Neumann 1940/50s) consider simplifying one component, and analyse its growth properties ψ n j = V n e iθj (45) here, θ is defined through grid size x and the total length L θ = 2π x L Consider simple Upwind method with σ = a t/ x (46) Substituting eq. (45) ψ n+1 j ψ n j + σ(ψ n j ψ n j 1 ) = 0 (47) V n+1 e iθj = V n e iθj + σv n [ e iθ(j 1) e iθj] dividing by V n and e iθj yields V n+1 V n ( ) = 1 + σ e iθ 1 (48) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

21 Stability annalysis II For the square of the absolute value one obtains λ(θ) V n+1 2 [ ( )] [ ( )] V n = 1 + σ e iθ σ e iθ 1 ( ) ( ) = 1 + σ e iθ + e iθ 2 σ 2 e iθ + e iθ 2 = 1 + σ(1 σ)(2 cos θ 2) ( ) θ = 1 4σ(1 σ) sin 2 2 (49) The method is now stable, if the magnitude of the amplification factor λ(θ) is smaller than unity. The upwind-method is stable for 0 < σ < 1, the λ(θ) < 1. Rewritten x t < f CFL (50) a with the Courant-faktor f CFL < 1. Typically f CFL = 0.5. Theorem: Courant-Friedrich-Levy There is no explizit, consistent and stable finite difference method which is unconditionally stable (i.e. for all t). W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

22 Modified Equation I Consider again Upwind method mit σ = a t/ x ψ n+1 j ψ n j + σ(ψ n j ψ n j 1 ) = 0 (51) substitute differences by derivatives, i.e. Taylor-series (up to 2. order) ( ψ t+1 ψ ψ 2 t t2 +O( t 3 )+σ x x 1 2 ) ψ 2 x 2 x 2 +O( t x 2 ) = 0 (52) divied by t, and substitute for σ ψ + a ψ x ( ψ t t a 2 ψ x 2 x ) + O( t 2 ) + O( x 2 ) = 0 (53) Use wave equation ψ tt = a 2 ψ xx modified equation (index M) ψ M + a ψ M x The FDE adds a new diffusive term to the original PDE = 1 2 a x (1 σ) 2 ψ M x 2 (54) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

23 Modified Equation II with the diffusion coefficient D = 1 a x (1 σ) (55) 2 Note: only for D > 0 this is a diffusion equation, and it follows σ < 1 for stability. (Hirt-method). For Upwind-Method D > 0 Diffusion. Lax-Wendroff yields ψ M The equation has the form + a ψ M x = t2 a σ ( ) σ 2 3 ψ 1 M x 3 (56) ψ t + aψ x = µψ xxx (57) µ = t 2 a ( σ 2 1 ) (58) σ This implies Dispersion. Here: waves are too slow (µ < 0) Oscillations behind the diskontinuity (cp. square function) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

24 The time step From the above analysis: the time step t has to be limited for a stable numerical evolution. For the linear Advection (with the velocity a) we find t < x (59) a In the more general case the sound speed has to be included and it follows the Courant-Friedrich-Lewy-condition t < x c s + u physically this means that information cannot travel in one timestep more than one gridcell. Typically one writes x t = f C c s + u with the Courant-Factor f C. For 2D situations: f C 0.5. Only for impliciten methods there are (theoretically) no limitations of t. (60) (61) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

25 Time step size - graphically The numerical region of dependence (dashed line) should be larger than the physical one (gray shaded area) since x/ t > a. The complete information from inside the physical sound cone should be considered. W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

26 Multi-dimensional Grid definition (in 2D, staggered): skalers in cell centers (hier: ρ, ɛ, p, v 3, ψ) Vectors ar interfaces (here: v 1, v 2 ) Fluxes across cell boundaries Top: mass flux bottom: X-momentum (griud shifted) from: ZEUS-2D: A radiation magnetohydrodynamics code for astrophysical flows in two space dimensions. I in The Astrophysical Journal Suppl., by Jim Stone and Mike Norman, Use Operator-Splitting and Directional Splitting: The x and y direction are dealt with subseuqently. First x-scans, then y-scans. W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

27 Summary: Numerics Numerical methods should resemble the conservation properties. write equations in conservative form Numerical Methods should resemble the wave properties. Shock-Capturing methods, Riemann-solver Numerical Methods should control discontinuites. need diffusion ( stability) either explicitly (artificial viscosity) or intrinsically (through method) Numerical methods should be accurate min. 2nd order in space and time Freely available codes: ZEUS: classical Upwind-Code, 2nd order, staggered grid, RMHD ATHENA: successo of ZEUS: Riemann solver, centered grid, MHD NIRVANA: 3D, AMR, finite volume code, MHD PLUTO: 3D, relativistic, Riemann-solver/finite volume, MHD GADGET: W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

28 Hydrodynamics: Wave structure Consider one-dimensional equtions (motion in x-direction): From Euler equations: With p = (γ 1)ρɛ and separation of derivatives ρ + ρu ρu ρɛ As Vector equation mit x = 0 + ρuu x = p x + ρɛu x = p u x W = ρ u p = W ρ + u ρ x + ρ u x = 0 u + u u x + 1 p ρ x = 0 p + u p u x + γp x = 0 + A W x = 0 (62) und A = Equations are non-linear and coupled. Try decoupling: Diagonalisation of A u ρ 0 0 u 1/ρ 0 γp u (63) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

29 Hydrodynamics: Diagonalisation Eigenvalues (EV) det(a) = It follows u λ ρ 0 0 u λ 1/ρ = (u λ) u λ 1/ρ 0 γp u λ γp u λ [ ] = (u λ) (u λ) 2 γp/ρ = 0 (64) λ 0 = u λ ± = u ± c s (65) with the sound speed cs 2 = γp ρ The Eigenvalues are the charakteristic velocities, with which the information is spreading. It is a combination of fluid velocity (u) and sound speed (c s) 3 real Eigenvalues A diagonalisable (66) Q 1 AQ = Λ (67) Q is built up from the Eigenvalues to the EV (λ i, i = 0, +, ), Λ is a diagonal matrix. W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

30 Hydrodynamics: Charakteristic Variables For Q it follows 1 ρ 1 2 c s 1 ρ c s Q = und Q 1 cs 2 1 = ρc s 1 2 ρc ρc s s ρc s We had and Define: W + A W x = 0 (68) Q 1 AQ = Λ dv Q 1 dw also dw = Qdv (69) Multiply eq. (68) with Q 1 v + Λ v x = 0 (70) v = (v 0, v +, v ) are the charakteristic Variables: v i = const. in curves dx dt = λ i W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

31 Hydrodynamics: The variable v 0 from the definitions dv 0 = dρ 1 c 2 s dp (71) v 0 + λ 0 v 0 x = 0 mit λ 0 = u (72) What is dv 0? From thermodynamics (1. Law) for specific quantities) ( ) 1 Tds = dɛ + p d = dɛ p ( ) 1 ρ ρ d 2 ρ with p = (γ 1)ρɛ, ɛ = c vt, γ = c p/c v it follows [ ds = cp dρ dp ] = cp ρ cs 2 ρ dv 0 (74) = s + u s x = 0 (75) i.e. s is const. along stream lines, hence ds dt (73) = 0 (76) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

32 Hydrodynamics: Riemann-Invariants For the additional variables with it follows v ± + (u ± c s) v± x = 0 (77) dv ± = du ± 1 ρc s dp (78) v ± = u ± Let the entropy constant everywhere (i.e. p = K ρ γ ) dp ρc s. (79) = v ± = u ± 2cs γ 1 (80) Riemann-Invariants: v ± = const. on curves dx dt = u ± c s W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

33 Hydrodynamics: Steepening of sound waves Linearisation of the Euler-equation resukts in the wave equation for the perturbations: ρ 1 = c 2 2 ρ 1 Example for (receeding) shock wave s (81) t x 2 but: c s is not constant steepening Diskontinuities W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

34 Examples: Shocktube Initial discontinuity in a tube at position x 0 (one-dimensional) Bereich 2 Riemann-Problem Bereich 1 Jump in pressure (p) and density (ρ) Diskontinuität P2 Evolution: - a shock wave to the right (X 4 ) ρ 2 (supersonically u sh > c s ) - a contact discontinuity ρ 1 P1 density jump (along X 3 ) - a rarefaction wave x (between X 1 and X 2 ) 0 W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

35 Examples: Sod-Shocktube A standard test problem for numerical hydrodynamics, x [0, 1] with X 0 = 0.5, γ = 1.4 ρ 1 = 1.0, p 1 = 1.0, ɛ 1 = 2.5, T 1 = 1 and ρ 2 = 0.1, p 2 = 0.125, ɛ 2 = 2.0, T 2 = 0.8 Hier solution with van Leer method (at time t = after 228 time steps:) Velocity Temperature Shock-Tube: Sod; Mono: Geometric Mean; Nx=100, Nt=228, dt= Density Pressure Red: Exact Green: Numerics The solution is: self similar obtained through stretching X-Axis X-Axis W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

36 Examples: Sedov-Explosion An example for bomb explosions (Sedov & Taylor, 1950s), analytical solution (Sedov) Basic setup for Supernovae-outbusts, e.g. estimate of the remnant size Standard test problem of multi-dimensional hydrodynamics, e.g. for x, y [0, 1] [0, 1] Energy-Input at origin, E = 1, in ρ = 1, γ = 1.4, grid points Here: solution with van Leer method (solve for total energy variable). Plotted: density W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

37 Examples: Water droplet: SPH Water sphere (R=30cm), Basin (1x1 m, 60cm high) Incl. surface tension, time in seconds (TU-München, 2002) (Website) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

38 Examples: Ster formation: SPH Molecular Cloud Mass: 50 M Diameter: 1.2 LJ = 76,000 AU Temperature: 10 K (M. Bate, 2002) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

39 Examples: Kelvin-Helmholtz Instability I W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

40 Examples: Kelvin-Helmholtz Instability II Direct comparision: moving < > fixed grid Left: Moving grid (Voronoi-Tesselation) Right: fixed square grid (Euler) with grid motion displayed (Kevin Schaal, Tübingen) Youtube channel W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

41 Examples: Kelvin-Helmholtz Instability III (Boulder (NCAR), USA) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

42 Examples: Rayleigh-Taylor Instability PPM Code, 128 Nodes, ASCI Blue-Pacific ID System at LLNL Grid Cells (LLNL, 1999) (Web-Link) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

43 Examples: Diesel Injection Finite Volumen Method (FOAM) Velocity, Temperature, Particles (+Isosurfaces) (Nabla Ltd, 2004) W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

44 Examples: Cataclysmic Variable: Grid y x W. Kley Numerical Hydrodynamics: A Primer Bad Honnef

45 Examples: Kataclysmic Variable: Disk formation RH2D Code, Van Leer slope Gridpoints, mass ratio: q = m 2 /m 1 = 0.15 W. Kley Numerical Hydrodynamics: A Primer Bad Honnef