Numerical Simulations. Duncan Christie

Transcription

1 Numerical Simulations Duncan Christie

2 Motivation There isn t enough time to derive the necessary methods to do numerical simulations, but there is enough time to survey what methods and codes are available and when they are used. If you now what is available and possible, you can go in search of more information Types of Simulations Outline N-Body Simulations Fluid Simulations Other Interesting Methods

3 Problems Non-linear (magneto-)hydrodynamics simulations Star formation Cosmological simulations of galaxies and cosmological structure Accretion Disks Exoplanetary atmospheres Stellar structure N-body Simulations Dark Matter Stellar clusters In the absence of gas Coupled to hydrodynamic simulations Planetary dynamics

4 N-Body Simulations The goal of an n-body simulation is to evolve a system of particles under influence of a force (usually gravity). This gives a deceptively simple system of equations: a i = dv i dt = a i G NX j=1,j6=i dx i dt = v i m j x j x i x j x i 3 Although simple, it is surprisingly difficult to maintain stability and energyconservation Codes: NBODY6 (Pure N-Body), GADGET (N-Body + SPH)

5 N-Body Example: The Millennium Simulation The Millennium project used 10 billion particles to simulation the evolution of dark matter in a box 2 light years on a side.

6 From N-Body to Hydrodynamics Why can t fluid dynamics be done in the same way? Consider two packets of gas moving towards each other in the absence of gravity: Fluids interact via pressure (collisions on the microscopic scale). Without it, the fluid packets pass through each other without interacting. How do we capture this?

7 From N-Body to Hydrodynamics Why can t fluid dynamics be done in the same way? Consider two packets of gas moving towards each other in the absence of gravity: Fluids interact via pressure (collisions on the microscopic scale). Without it, the fluid packets pass through each other without interacting. How do we capture this? Option 1: Model pressure interactions between gas particles using smooth particle hydrodynamics. Option 2: Treat particles as representative of actual particles and model interactions as collisions using direct simulation Monte Carlo (DSMC). Option 3: Give up on thinking about particles and think about density and velocity fields living on a grid instead.

8 From N-Body to Hydrodynamics Why can t fluid dynamics be done in the same way? Consider two packets of gas moving towards each other in the absence of gravity: Fluids interact via pressure (collisions on the microscopic scale). Without it, the fluid packets pass through each other without interacting. How do we capture this? Option 1: Model pressure interactions between gas particles using smooth particle hydrodynamics. Too expensive in most cases! Option 2: Treat particles as representative of actual particles and model interactions as collisions using direct simulation Monte Carlo (DSMC). Option 3: Give up on thinking about particles and think about density and velocity fields living on a grid instead.

9 Equations of (Magneto-)Hydrodynamics Continuity Equation Momentum Equation Conservation + r ( v) =0 Energy Equation + r ( vv) = rp + g (r B) B Energy Update: How the gas heats + r [(E + p? ) v B (B v)] = v g + Induction Equation Poisson equation Magnetic r 2 r (v B) =0 Gravitational Field =4 G Flux Freezing

10 Smoothed-Particle Hydrodynamics In SPH, the fluid is modeled as a system of particles. It is different from n-body simulations in that there are interactions with other particles taken into account to model things like pressure. A smoothing kernel is applied which weights closer neighbors more heavily. The kernel has a characteristic length h, and particles outside a few h are safely ignored.

11 Smoothed-Particle Hydrodynamics In SPH, the fluid is modeled as a system of particles. It is different from n-body simulations in that there are interactions with other particles taken into account to model things like pressure. A smoothing kernel is applied which weights closer neighbors more heavily. The kernel has a characteristic length h, and particles outside a few h are safely ignored. h Close Neighbor Weighted Heavily

12 Smoothed-Particle Hydrodynamics In SPH, the fluid is modeled as a system of particles. It is different from n-body simulations in that there are interactions with other particles taken into account to model things like pressure. A smoothing kernel is applied which weights closer neighbors more heavily. The kernel has a characteristic length h, and particles outside a few h are safely ignored. h 2h Close Neighbor Weighted Heavily Included in the calculation, but with a smaller weight

13 Smoothed-Particle Hydrodynamics In SPH, the fluid is modeled as a system of particles. It is different from n-body simulations in that there are interactions with other particles taken into account to model things like pressure. A smoothing kernel is applied which weights closer neighbors more heavily. The kernel has a characteristic length h, and particles outside a few h are safely ignored. h 2h Close Neighbor Weighted Heavily Included in the calculation, but with a smaller weight Too far away, so safely ignored.

14 Smoothed-Particle Hydrodynamics Advantages SPH particles naturally accumulate where resolution is needed (resolution is limited by the smoothing length and the number of particles) Fairly simple to implement Disadvantages Additional physics can be difficult to add to SPH compared to grid-based codes (although it can be done). Not great at resolving shocks, although this is improving. Common SPH Codes GADGET (and it s derivatives) HYDRA GASOLINE NDSPMHD (David Price s SPH MHD code has some interesting features but a terrible name)

15 Grid-based Codes Grid-based codes work by discretizing the field variables and placing them on a grid. Grid with all variables cell-centered Ignoring source terms (gravity, etc.), the hydrodynamic equations can be + r F =0 i-1,j,k,v1,v,b1,b2,e i,j,k i+1,j,k where q is the fluid variable and F is the associated flux. In 1-d, we can discretize this as Grid with vector components staggered r F = 1 x F i+1/2,j,k F i 1,j,k The time derivative also needs to be = 1 t q n+1 q n i-1,j,k i,j,k v2,b2 i+1,j,k

16 Grid-based Codes Combining all of these results, we get q n+1 i,j,k = qn i,j,k + t x F n+1/2 i+1,j,k F n+1/2 i 1,j,k Thus, to advance from t n to t n+1 we need to specify how we get the timecentered/time-averaged fluxes F from the fluid variables at time n (q n ). The prescription that gives you the fluxes depends on the solver and will impact how accurate the solution is. Godunov Scheme: Reconstruct the fields (as constants, linear functions, second order functions, etc.) to get values on the boundaries. This yields a discontinuous interface and initial values for the Riemann Problem. Solving the Riemann problem yields the fluxes.

17 Time stepping and the Courant Condition In advancing the solution from step n to step n+1, how large can t n+1 -t n be? In general, when you have a wave with a specified signal speed (like the speed of sound), the solution can be advanced more than the signal crossing time of a single zone, t< x c s The Courant number is the scaling that determines the actual time step C = c s t x A typical value of C is 0.3 to 0.5 which is chosen ensure that the time step isn t too large.

18 Time stepping and the Courant Condition Diffusive processes complicate measures as they don t have a finite signal speed. The stability condition is t<(constants) x 2 where the constants depend on the physical process in question. This is a rather strict requirement and becomes costly when attempting highresolution simulations.

19 Adaptive Mesh Refinement (AMR) It may be expensive to use a fine grid to resolve the entire simulation domain. Most grid-based codes allow for finer grids to overlay coarser grids, allowing for objects of interest to be targeted. Coarse Grid Conditions can be chosen to add additional grids to resolve specific length scales. Truelove (1998) asserts that the Jeans length must be resolved by at least 4 zones.

20 Adaptive Mesh Refinement (AMR) It may be expensive to use a fine grid to resolve the entire simulation domain. Most grid-based codes allow for finer grids to overlay coarser grids, allowing for objects of interest to be targeted. Coarse Grid + 1 Level of AMR Conditions can be chosen to add additional grids to resolve specific length scales. Truelove (1998) asserts that the Jeans length must be resolved by at least 4 zones.

21 Adaptive Mesh Refinement (AMR) It may be expensive to use a fine grid to resolve the entire simulation domain. Most grid-based codes allow for finer grids to overlay coarser grids, allowing for objects of interest to be targeted. Coarse Grid + 2 Levels of AMR Conditions can be chosen to add additional grids to resolve specific length scales. Truelove (1998) asserts that the Jeans length must be resolved by at least 4 zones.

22 Resolving Relevant Physical Scales Here are two simulations of protoplanetary disks. Due to the lack of resolution, unstable modes are stabilized, preventing fragmentation. Low Resolution High Resolution

23 Resolving Relevant Physical Scales Here are two simulations of protoplanetary disks. Due to the lack of resolution, unstable modes are stabilized, preventing fragmentation. Takeaway: Be aware of the implications of unresolved length scales. Low Resolution High Resolution

24 Common Grid-based Codes ZEUS: A grid based code originally developed in It has been through many iterations (ZEUS-2D, ZEUS-3D, ZEUS-MP, ZEUS-MP/2, etc.). It is fast and easy to use; however, it doesn t support AMR and is very diffusive. ENZO and RAMSES: Two modern codes. Enzo is used by a number of people in the department (myself included). FLASH: Developed by the Flash Center for Computational Science. Has been used for simulations of everything from laser experiments to star formation in the ISM. ATHENA: Developed to be a successor to ZEUS.

25 A Random Selection of Other Methods

26 Direct Simulation Monte Carlo (DSMC) Although particle approaches to fluid dynamics a prohibitive, there are times when they are necessary, specifically when the mean free path is comparable to or larger than the relevant scales in the problem or when the particles don t obey a Boltzmann distribution. The Knudsen number is the ratio of the mean free path to the relevant length scale. Kn = When Kn is larger than of approaches 1, fluid approximations break down. Discrete Simulation Monte Carlo (DSMC) is a method for simulation collisional fluids using pseudo-particles. It is very expensive but it is the only way to capture gas behavior when the fluid approximation breaks down. An astrophysical application: Modeling the upper atmospheres of planetary bodies like Pluto and Titan. Both are losing their atmospheres but the atmospheres are diffuse enough that the upper atmospheres can t be treated like fluids. mfp L

27 Particle-In-Cell The MHD equations are formulated assuming the gas obeys a Boltzmann distributions. Particle-in-cell codes sample the underlying distribution (Boltzmann, for example) then updates the particle positions and velocities. The new distribution is then determined from the updated particle velocities.