Numerical Astrophysics: hydrodynamics

Transcription

1 Numerical Astrophysics: hydrodyamics

2 Part 1: Numerical solutios to the Euler Equatios

3 Outlie Numerical eperimets are a valuable tool to study astrophysical objects (where we ca rarely do direct eperimets). Numerical simulatios i astrophysics cover a wide rage of situatios, with may techiques ad various degrees of compleity. The most widely used i astrophysics ca be roughly divided ito: N-Body dyamics; Fluid dyamics (hydrodyamics, magetohydrodyamics, SPH...). This first series of lectures will provide a hads o approach to umerical hydrodyamics. The goal is to write a series of codes to solve the equatios of gas dyamics i oe dimesio, or i multi-dimesios. Material (templates ad sample codes) will be made available i fortra 90, as well as basic plottig scripts i guplot or pytho (matplotlib).

4 Hydrodyamics I order to describe a system of particles as a fluid several coditios must be met The mea free path has to be smaller tha the scale-legth of the fluctuatios of the macroscopic variables (such as desity, pressure...). mfp L The time betwee collisios must be small compared with the time-scale of chages i the flow. t coll t flow The mea distace betwee particles has to be smaller tha the scale-legth of the macroscopic variables. l = 1/3 L

5 The Euler equatios We cosider a series of fied volume elemets (Euleria mesh). The equatios that describe the fluid dyamics i a fied volume elemet (disregardig viscous effects) are: Mass coservatio + r ( u) Mometum (Newto s 2d u) + r ( uu)+rp = f Eergy + r [u (E + P )] = L + f et u Alog with a equatio of state of 55 PDE E = 1 2 u2 + P 1 they form a system

6 Cartesia coordiates, writte i matri @z = S U = u v w E 3 7 5, F = u u 2 + P uv uw u(e + P ) 3 7 5, G = v vu v 2 + P vw v(e + P ) 3 7 5, H = w wu wv w 2 + P w(e + P ) 3 7 5, S = G 0 f f y f z L + f u ad u=(u, v, w) coserved variables,y, ad z flues source terms

7 Fiite differeces Discretize the coordiate i N poits i = 0 + i, i =1, 2,...N cosider a fuctio f()! f( i )=f i the derivative of f() ca be approimated by fiite differeces f 0 fwd() = f 0 back() = f( + ) f() f() f( ) = f i+1 f i = f i f i 1 f f 0 cet( i ) f i+1 f i 1 2 f i f i+1 f() f i 1 f 0 cet() = f( + ) f( ) 2 = f i+1 f i 1 2 Remember the formal defiitio of the derivative f 0 () = lim h!0 f( + h) f() h i 1 i i+1

8 Applicatio of fiite differeces: FTCS Discretize space as: i = 0 + i ; i =1, 2... Ad time as: t = t 0 + t;,=1, 2,... With this otatio U(, t) =U i The we apply forward differece i time (FT), ad cetered differecig i U+1 i t F i+1 F i @ = S ) U+1 i = U i t 2 F i+1 F i 1 + ts However, it turs out to be ucoditioally ustable for hyperbolic equatios (i.e. useless for our case)

9 The La-Friedrichs method La fied the problem of stability of the FTCS by replacig U i! 1 2 U i+1 + U i 1 Thus the La Method reads: U +1 i = 1 2 U i+1 + U i 1 t 2 F i+1 F i 1 + ts However, we ca ot take arbitrarily large values of restricted by a stability criterium. t, they are

10 The advectio equatio Cosider the liear + =0 with iitial coditios has ) = S U = u, F = au, ad S =0 u(, t = 0) = f() u(, t) =f( at) f a<0 f(, t = 0) a>0

11 Vo Neuma aalysis Cosider the advectio equatio If oe assumes that oe has: Thus the error follows Apply La Method to oe Fourier mode ) +1 i = 1 2 ˆ +1 i+1 + i 1 + a @t + =0 u =ū + i+1 i 1 true solutio ˆ = 1 2 ejk +e jk + a t 2 ejk +e jk error(e.g. trucatio) (, t) = i = ˆ (t)e jk = ˆ e jk ; j = p 1 = cos(k) j a t si(k) The error grows if ˆ +1 ˆ > 1 The coditio for stability becomes a t = a t = Co apple 1 This is the famous Courat-Friedrichs-Lewy stability coditio

12 Geeral solutio strategy (1D): 1. Discretize domai i N (+2) poits i =0, 1, 2...N +(N + 1) 2. Iitial coditios i the etire domai: 3. Determie the timestep allowed by the CFL coditio t = C,withC<1 a 4. Calculate the Flues F i = F(U i ) 5. Advace the solutio, e.g.: U +1 i = 1 2 U i+1 + U i 1 t 2 6. Apply boudary coditios U +1 0, U +1 N+1 F i+1 F i At this poit = +1, (t = t + t) ad we iterate ts

13 Eample of the advectio equatio solved with the La method Advectio of a regular pulse Periodic Boudary coditios: U +1 0 = U +1 N, U +1 N +1 = U+1 1 i =0 i =1 i =2... i = Ni =(N+ 1) v Advectio eq. w/la method t= 0 t t= 2 t t= 4 t t= 6 t Eample code i fortra: advectio_la.f90 Guplot plottig script plot_advectio.gpl pytho (matpotlib) plottig script plot_advectio.py Dowload from: Click o the right o the tab: Hydrodyamics ICTP-SAIFR School

14 Pgas Task 1: Write a program (suggestio modify the advectio_la.f90) to solve the Euler equatios. Iitial coditios: classic Sod tube test Euler equatios w/la s Method, t= L =1.0 P L = u R =0.1 u L = u R =0 0.0 P R = You will eed to chage u(0:+1)to u(3,0:+1) u(1,i):desity at _i u(2,i): mometum at _i u(3,i): total eergy desity (same with F, UP) Set the iitial coditios (see fig) Chage output to write desity, velocity, ad Pressure The ew CFL coditio is Co t =mi u + cs Boudary coditios: ouflow Itegrate oly to all domai U +1 0 = U +1 1, U +1 N +1 = U+1 N t ma =0.2

15 Here s how the Sod tube test solutio looks like after t=0.2 Euler equatios w/la s Method, t=0.2 Rarefactio wave u Pgas cotact discotiuity Shock