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### class19

Course: ASTR 415, Fall 2008
School: Maryland
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Word Count: 671

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19. Class N -body Techniques, Part 2 Time-integration Schemes Clearly, Newtons laws are IVP. Could use any method (Euler, RK4, etc.). But, issue is to balance accuracy vs. eciency. Typically need many particles to capture dynamics correctly (e.g., in stellar system or galaxy). This consideration may be as important as accuracy of any one individual particle (exception: solar systemN 9, 109 1010 orbits)....

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19. Class N -body Techniques, Part 2 Time-integration Schemes Clearly, Newtons laws are IVP. Could use any method (Euler, RK4, etc.). But, issue is to balance accuracy vs. eciency. Typically need many particles to capture dynamics correctly (e.g., in stellar system or galaxy). This consideration may be as important as accuracy of any one individual particle (exception: solar systemN 9, 109 1010 orbits). Could use Euler scheme. But we have seen it is just as easy to design 2nd -order scheme by centering derivatives = could use leapfrog (very stable). Practical timestep control The stability criterion from the discussion of sti systems also applies to the leapfrog integrator for the N-body problem. Can show need t < 2/, where 2 = | F| is a characteristic interaction frequency for a particle (in practice, need t 2/ to avoid problems). But 2 is dierent for every particle; can be very large for particle undergoing close interaction. If have to take max , can be very restrictive. Two solutions: 1. Use dierent timesteps for each particle (individual timesteps). E.g., ti = Fi /Fi eective, but complex implementation, and may break symplecticity of leapfrog (for example). More complex expressions for ti can be formulated, e.g., involving higherorder derivatives of F . These are largely heuristics with convenient properties. It is dicult to prove analytically that one formulation is superior to another. Sometimes ti is discretized, e.g., in factors of 2 (multistepping). 2. Eliminate short-timescale phenomena by modifying gravity on small scales. E.g., Set t = D /30 and/or use softening (see below). Always important to check whether simulation is giving physically meaningful results. Handy technique: reduce timestep by factor of 2 to see if global behavior strongly aected. If so, may have to use smaller steps. Beware of chaos: if state of system strongly dependent on initial conditions, change of timestep may give seemingly vastly dierent results. Need to monitor constants of motion to be sure. 1 Force evaluation Solving the IVP requires evaluation of the RHS of the ODEs, i.e., must compute interparticle forces. discuss Will PP, PM, P3 M, and tree methods. But rst must consider another practical issue, related to timestep control... Hard interactions Recall F ij = Gmj (ri rj )/|ri rj |3 . Problem: if |ri rj | is small, |F ij | diverges, leading to timestep trouble as |vi | . Physically, very close encounters occur on very short timescales, e.g., can form close binaries with very short periods. To alleviate problem, could use softened forces: F ij = where = softening parameter. Maximum force now Gm2 /2 . Physically, this eliminates possibility of forming binaries with r < . OK when particles represent collection of stars on similar orbits. Not OK if studying small clusters, where each particle represents an individual star. In this case binaries can form and signicantly aect evolution of entire cluster. Modern methods also sometimes use regularization. Binaries (or hierarchies) replaced by pseudo-particles until interaction with other particles becomes important. Gmj (ri rj ) , (|ri rj |2 + 2 )3/2 Direct Summation (PP Method) Most straightforward way of evaluating Fij . 1 But number of operations = 2 N(N 1) N 2 for N that Fij = Fji . 1 (the 1 2 comes from the fact 10 more particles = 100 more work. Severely limits number of particles that can be...

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Maryland - ASTR - 20
Class 20. N -body Techniques, Part 3The PM Method, ContinuedThere are several distinct steps in PM process: 1. Assign particles to mesh to compute i . 2. Get boundary conditions for (0 and N +1 ). 3. Solve discretized version of Poissons equation.
Maryland - ASTR - 415
Class 20. N -body Techniques, Part 3The PM Method, ContinuedThere are several distinct steps in PM process: 1. Assign particles to mesh to compute i . 2. Get boundary conditions for (0 and N +1 ). 3. Solve discretized version of Poissons equation.
Maryland - ASTR - 21
Class 21. N -body Techniques, Part 4Tree CodesEciency can be increased by grouping particles together: Nearest particles exert greatest forces direct summation. Distant particles exert smallest forces treat in groups.Treat distant particles as
Maryland - ASTR - 415
Class 21. N -body Techniques, Part 4Tree CodesEciency can be increased by grouping particles together: Nearest particles exert greatest forces direct summation. Distant particles exert smallest forces treat in groups.Treat distant particles as
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Class 22. PDEs, Part 1 Cf. NRiC 19.Classication of PDEs A PDE is simply a dierential equation of more than one variable (so an ODE is a special case of a PDE). PDEs are usually classied into three types: 1. Hyperbolic (second or rst order in time
Maryland - ASTR - 415
Class 22. PDEs, Part 1 Cf. NRiC 19.Classication of PDEs A PDE is simply a dierential equation of more than one variable (so an ODE is a special case of a PDE). PDEs are usually classied into three types: 1. Hyperbolic (second or rst order in time
Maryland - ASTR - 23
Class 23. PDEs, Part 2Solving Hyperbolic PDEs, ContinuedUpwind dierencing In addition to amplitude errors (instability or damping), scheme may also have phase errors (dispersion) or transport errors (spurious transport of information). Upwind die
Maryland - ASTR - 415
Class 23. PDEs, Part 2Solving Hyperbolic PDEs, ContinuedUpwind dierencing In addition to amplitude errors (instability or damping), scheme may also have phase errors (dispersion) or transport errors (spurious transport of information). Upwind die
Maryland - ASTR - 24
Class 24. Fluid Dynamics, Part 1 The equations of uid dynamics are coupled PDEs that form an IVP (hyperbolic). Use the techniques described so far, plus additions.Fluid Dynamics in Astrophysics Whenever mean free path problem scale L in a plasm
Maryland - ASTR - 415
Class 24. Fluid Dynamics, Part 1 The equations of uid dynamics are coupled PDEs that form an IVP (hyperbolic). Use the techniques described so far, plus additions.Fluid Dynamics in Astrophysics Whenever mean free path problem scale L in a plasm
Maryland - ASTR - 415
ASTR415 Survey ResultsSpring 2007 11 respondents 1. Computer familiarity [1=Master, 5=None]: Avg = 2.7 [Skilled], Min = 1 [Master], Max = 4 [Novice] 2. Unix familiarity [1=Master, 5=None]: Avg = 3.6 [Skilled], Min = 2 [Expert], Max = 5 [None] 3. Uni
Maryland - ASTR - 415
ASTR415 Spring 2007Due May 08, 2007Term Project1) For your term project you will install and learn how to use a freely available (opensource) 3D visualization tool based on Open-GL. You will write a short report and present in class the results
Maryland - ASTR - 415
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Maryland - ASTR - 415
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Maryland - ASTR - 415
4Xeex xe e7jj \$v )e `j B I Va U b V V U R I F YH Y W V U W R H c W V V F Y WH Q V I Pgp1eCd)wss`Xgpwhe`Xe1wPX)1T Ia R Y W R S U b V Y I W R U U R S Y I R U V F Y I V H b U U T I Va b Y r WH Raa Rt V F D
Maryland - ASTR - 415
ICadafWWaHH\$CHIaRp\$a\$wraWCawHa 9 i i S x i U bIxpU B @ e F k i 6 d B wP v q D A B FP PT B 8 6 qT Q v D 4 B v 4 @ D B F f D F B A @ A B F 6P k Uif'fbS7)UdRT Y rtuCaCdRR99R7\$17t\$ICdP Y `CaC9ECGR'\$i S
Maryland - ASTR - 415
A Crash Course on UNIXUNIX is an &quot;operating system&quot;.Interface between user and data stored on computer. A Windows-style interface is not required. Many flavors of UNIX (and windows interfaces).Solaris, Mandrake, RedHat (fvwm, Gnome, KDE)
Maryland - ASTR - 415
Data RepresentationsComputers store data as different variable types, e.g. integer, floating point, complex, etc. Different machines have different wordlengths, e.g. 4-byte ints on a 32-bit machine (Pentium), 8-byte ints on a 64-bit machine (Alpha).
Maryland - ASTR - 415
VisualizationVisualization is useful for:1) Data 2) Codeentry (initial conditions) debugging and performance analysis and display of results3) Interpretation Our focus will be #3. The computational astrophysicist can either:1) Develop 2) U
Maryland - ASTR - 415
Numerical Linear Algebra Probably the simplest kind of problem. Occurs in many contexts, often as part of larger problem. Symbolic manipulation packages can do linear algebra &quot;analytically&quot; (e.g. Mathematica, Maple). Numerical methods needed when:
Maryland - ASTR - 415
Nonlinear EquationsOften (most of the time?) the relevant system of equations is not linear in the unknowns. Then, cannot decompose as Ax = b. Oh well. Instead write as:(1) (2) f(x) = 0 f(x) = 0function of one variable (1-D) x = (x1,x2,.,xn
Maryland - ASTR - 415
Statistical Description of Data Cf. NRiC, Chapter 14. Statistics provides tools for understanding data.In the wrong hands these tools can be dangerous! Apply some formula to data to compute a &quot;statistic&quot;. Find where value falls in a probability
Maryland - ASTR - 415
Modeling of Data NRiC Chapter 15. Model depends on adjustable parameters. Can be used for &quot;constrained interpolation&quot;. Basic approach:1. 2. 3. 4.Choose figure-of-merit function (e.g. 2). Adjust best-fit parameters: minimize merit function. Co
Maryland - ASTR - 415
Random Numbers NRiC Chapter 7. Frequently needed to generate initial conditions. Often used to solve problems statistically. How can a computer generate a random number? It can't! Generators are pseudo-random. Generators are deterministic: i
Maryland - ASTR - 415
Numerical Integration (Quadrature) NRiC Chapter 4. Already seen Monte Carlo integration. Can cast problem as a differential equation (DE): = is equivalent to solving for I y(b) the DE dy/dx = f(x) with the boundary condition (BC) y(a) =
Maryland - ASTR - 415
Ordinary Differential Equations (ODEs) NRiC Chapter 16. ODEs involve derivatives wrt one independent variable, e.g. time t. ODEs can always be reduced to a set of firstorder equations (involving only first derivatives).e.g. = is
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MATH 241 CALCULUS III FIRST MIDTERM EXAM SOLUTIONS (1) For this problem, u = + k, and v = + 2 + 3 k. ij i j (a) u v = 1 + 2 3 = 2. k ij (b) u v = det 1 1 1 = 5 2 + 3 k. i j 1 2 3 (c) The symmetric form of the equations are: 2 x =
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Maryland - ASTR - 120
Homework #2Chapter 3 Q25 The dividing line between the illuminated and unilluminated halves of the Moon is called the terminator. The terminator appears curved when there is a crescent or gibbous moon, but appears straight when there is a first qu
Maryland - ASTR - 120
ASTR120 Homework #3 (Hamilton) due Thursday Sept. 27 (20 Points)Finish reading Chapter 4! Now you are cleared for this homework (from Chapter 3, page 92). Try to start early if you can - there are some interesting (but tricky!) problems this week.
Maryland - ASTR - 120
ASTR120 Homework #4 (Hamilton) due Thursday Oct. 4 (30 Points)Denitely nish reading Chapters 4 and 7! These rst ve problems are from Chapter 4. W4. Do problem W4 from http:/www.astro.umd.edu/hamilton/ASTR120/webexp.html. 46. A satellite is said to
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Maryland - ASTR - 120
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Maryland - ASTR - 120
ASTR120 Homework #7 (Hamilton) due Thursday Nov. 1 (20 Points)Finish reading Chapters 10 and 11. These problems are from Chapter 10. 29. Temperature variations between day and night are much more severe on the Moon than on Earth. Explain why. 31. U
Maryland - ASTR - 120
ASTR120 Homework #8 (Hamilton) due Thursday Nov. 8 (20 Points)Finish reading Chapters 11 and 12! This is the last graded homework before Exam 2 on Nov. 15. These problems are from Chapter 11. 56. The Mariner 2 spacecraft detected more microwave rad
Maryland - ASTR - 120
ASTR120 Homework #9 (Hamilton) due Thursday Nov. 29 (20 Points)Finish reading Chapter 12 and 13! These problems are from Chapter 12. 49. When Saturn is at dierent points in its orbit, we see dierent aspects of its rings because the planet has a 27
Maryland - ASTR - 120
ASTR120 Homework #10 (Hamilton) due Thursday Dec. 6 (20 Points)Finish reading Chapters 14, 15, and 28! These problems are from Chapter 14. 30. At certain points in its orbit, a stellar occultation by Uranus would not reveal the existence of the rin
Maryland - ASTR - 120
ASTR120 Challenge Problem #1 (Hamilton) Optional, due before Midterm #1In this challenge problem you will work out how large a shadow moons in our Solar System cast on their parent planets. a) Draw an accurate picture and use algebra and geometry t
Maryland - ASTR - 120
ASTR120 Challenge Problem #2 (Hamilton) Optional, due before Midterm #2In this challenge problem you will work out how Mars tilt and eccentricity aect its seasons. a) For an untilted Earth on a circular orbit at 1 AU, work out the average energy hi
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