093983 - 39/83 Semester 2, 2009 Page 1 of 8 THE UNIVERSITY...

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Unformatted text preview: 39/83 Semester 2, 2009 Page 1 of 8 THE UNIVERSITY OF SYDNEY FACULTIES OF ARTS, EDUCATION, ENGINEERING AND * SCIENCE COSC 3011 . SCIENTIFIC COMPUTING COSC 3911 - SCIENTIFIC COMPUTING (ADV) NOVEMBER 2009 Time allowed: TWOHOURS MARKS FOR QUESTIONS ARE AS INDICATED TOTAL: 60 MARKS INSTRUCTIONS o COSC 3011 students are to answer Questions 1, 2, 3, and 4 (3011) o COSC 3911 students are to answer Questions 1, 2, 3, and 5 (3911) o All answers should include explanations of reasoning 0 Where a question says to provide a brief explanation, write less than half a page (excluding diagrams) o Page 2 of this booklet lists some formulae which may prove useful 39/83 ‘ I Semester 2, 2009 Page 2 of 8 Formulae 1. Forward time (FT) approximation to the time derivative of f (3:, t) af_f(x>t+T)—"f(xit) ‘ 57_—~—-—7~———+0(T) 2. Centred time (CT) approximation to the time derivative of f (:13, t) fl_f(x,t+T)—f(w,t-T) 2 6% ~ 27' + 0(T ) 3. Centred space (CS) approximation to the spatial derivative of f (as, t) g: z f<z+h,t) ~f<x—h,t> 2 8x 2h + O<h ) 4. Centred space (CS) approximation to the second spatial derivative of f (:10, t) 82f __ hut) _ 21:03:?» + —— h)t) __ _ w 0 h2 8ch h2 + ( ) 5. Taylor series expansion of f (t) df 1 2612f 3 f(t+T)-—f(t)+7'%+§7' ) 6. Series approximation to — ln(1 — 3:) for < 1 1 l — ln(1 — as) 2 at + 2—552 + 3-133 + 0(5134) 6. Binomial approximation to (1 + :3)" for < l —— l (1 + m)“ = 1 + out; + 3W7~lx2 -|~ 0(273) 7. Geometric sum 2 N 1 TN ‘1 SN=1+T+T +r3+...+7"" :2 1 T _ 39/83 Semester 2, 2009 Page 3 of 8 Question 1 (3011 & 3911) [Question 1 continues over the page] (a) Briefly explain what is meant by the term local truncation error in the context of the numer— ical solution of a 1—D Ordinary Differential Equation (ODE) gm dt“ including briefly explaining the notation 0(7'k). flw(t),tl, (1) (b) Dynamics problems of the form: fizv and d—V—za (2) dt dt where r :2 r(t), v = V(t), and where a : a(r) is a given function, may be solved using the Velocity—Verlet method: rn+1 = rn + TVn + %’T‘2&n . (3) Vn+1 : Vn + %7' (an + an+1) ) (4) Where rm 2 r(tn), vn : V020, and tn 2 (n — 1)7". Show that the Velocity-Verlet scheme is exactly time reversible. (c) The ODE dgu du —-—— = —2d— - - 5 dt2 dt. “ ( ) is ‘stiff’ for d >> 1. An appropriate scheme to solve stiff problems is the ‘backwards Euler’ method, which in application to a system of ODEs dx —— = f t t 6 may be written: X(t+7*) =X(t)+rf[x(t+r),t+r], (7) where 7' is the time step. (i) Apply the backwards Euler method to Eq. (5) to obtain a pair of implicit equations for two dependent variables. (ii) Analytically solve the equations obtained in (c) (i) to obtain explicit forms suitable for numerical solution of Eq. (5). (d) Runge—Kutta methods are a powerful class of methods for solving ODEs which match Taylor series expansions to a given order. (i) Show that a Taylor series solution to the 1—D ODE Eq. (1) may be written as :c(t + 7') = 33(t) + Tf[:1:(t),t] + %T2 + + 0(T3). (8) 39/83 ‘ _ Semester 2, 2009 . Page 4 of 8 Question 1 (3011 & 3911) [Continued from previous page] (ii) By considering the two—variable Taylor series expansion 8 f(x + (555$ + 6t) 2 f(:E, t) + 65125;: + 623%; + O[(6-a:)2] + O[(5t)2], (9) with the choices 63: = T f and 5t 2 T, use Eq. (8) to derive a second order Runge~Kutta scheme. (15 marks) 39/83 Semester 2, 2009 Page 5 Of 8 Question 2 (3011 & 3911) The non—dimensional form of the 1—D time-independent Schrodinger equation is ' “—— + VOW/J = EQ/J, (10) where 10 = 1/)(30) is the wave function, V(:13) is a prescribed potential, and E is the energy of the quantum particle. (a) Consider a discretisation of the independent variable: :01- : a31+ih with h 2 (mL —— rug/(L m l) where t: 1,2, ...,L, (11) with corresponding dependent variable values 2p)- = Applying the centred difference approximation to the second derivative in (10) and assuming Dirichlet boundary conditions $0 = 7/)(581 ~ 2 O and 1/1.” = 1/)(331; + = 0, (12) show that Eq. (10) may be formulated as a matrix eigenvalue problem: Aw = E¢ with ~ 7/) =(¢1,¢2,...,¢L)T and A = ——21—D + v. (13) In your answer, explicitly identify the matrices D and V. (b) Briefly explain how the eigenvalue problem defined by Eq. (13) may be solved using nor— malised power iteration to obtain the eigenvector representing thesmallest energy wave func— tion, for a given choice of the potential (c) For a particular choice of the potential V(a:), the exact solutions to Eq. (10) have energies Ej =j2 where j=l,2,.... (14) Consider the application of over—relaxation to the matrix equation (13) for this potential with the iteration scheme ¢lm+1l = cqplml with C = (1 — w)1+ wA‘l, (15) where I is the identity matrix, and where m enumerates iterations. (i) Determine the eigenvalues Xj (w) for the matrix C, and hence explain why the iteration scheme (15) diverges for w > 2. (ii) Determine the optimal choice of w for this problem (for convergence to the wave func— tion corresponding to E1), for a general choice of initial wave function. (15 marks) 39/83 ' Semester 2, 2009 Page 6 of 8 Question 3 (3011 & 3911) The linear advection equation is 80 = —c§g where c > O is constant. I (16) 57; 8:2: This equation may be solved with the ‘leap—frog’ scheme: 1 CT ware—ail). 07> a} (a) Derive the leap—frog scheme by taking appropriate differencing of Eq. (16). What is the order of this scheme? (b) Draw a stencil representing this scheme, and briefly explain what is required to march this scheme forward in time. (c) Apply von N eumann stability analysis to Eq. (17) and show that the amplification factor is f=—ifsinkhj:x/1—f2sin2kh with f=%t, (18) and hence determine when the method is stable. (Note that you need to consider the cases f > 1 and f S 1 separately.) (d) Briefly explain whether the matrix approach to stability analysis introduced in the unit is applicable in this case, and if not, why not? (6) Consider the choice 7- : h/c in the scheme (17). Given starting values a? and a; which are exact for all j, does the scheme then provide an exact solution for n = 2, 3, ...? (15 marks) 39/83 Semester 2, 2009 Page 7 of 8 Answer Question 4 (3011) OR Question 5 (3911). Do NOT answer both questions. Question 4 (3011) The 2—D diffusion equation is 2 2 .8; = K, + where [‘5 > O is constant. ~ (19) This equation may be solved with the forward—time centred—space scheme with an equal spatial grid spacing h in :1: and y, and with a time step 7': HT 32‘ (a) Briefly explain how to implement the scheme (20) in a matrix formulation involving a vector T” of dependent variables at time tn, given that the problem involves two spatial dimensions. (You are not required to identify the matrix involved.) (b) Apply von Neumann stability analysis to Eq. (17) by looking for trial solutions of the form 727:“: 53+ (streamer—weave; J?)- <20) fl = '“(emjh + elkyjh), and show that the amplification factor is 2 g: l+%(coskmh+coskyh—2). (21) Hence determine when the method is stable. (c) The parabolic PDE (19) and the discretisation (20) may be used to solve the Laplace equation 82T 822/" _._ W = 0, 22 8532 + (93/2 ( ) a procedure called Jacobi relaxation. Briefly explain the appropriate choice of 'r in that case, and hence qualitatively explain why the total computation time for Jacobi relaxation scales as L4, where L is the number of spatial grid points in each dimension. (d) Jacobi relaxation may be represented by an iteration scheme with matrix formulation Tn+1 : GTn, Where T" is a vector of dependent variable values at iteration n. Given that the spectral radius of the matrix G is 71.2 provide a quantitative estimate of the rate of convergence of the iteration, i.e. an estimate of the number of iterations required to reduce unwanted terms in the sum resulting from iteration of Eq. (23) by a factor 10—1”. (15 marks) 39/83 Semester 2, 2009 ' Page 8 of 8 Answer Question 4 (3011) OR Question 5 (3911). Do NOT answer both questions. Question 5 (3911) The discrete Fourier transform of of a 2—D function f (:12, y) is L L fmn = Zijl exp —~ 1)(j — l) —— — 1)(l — 1)] (25) j=1 1:1 and the inverse transform is 1 L L N fjl : E Z menexp [git-Mm ~1)(jr— 1) + Egan — 1)(l — 1)], (26) 71121 n=1 where the function is given at discrete points (133- ,4”) with j = 1, 2, ..., L and l = l, 2, ..., L and Where sz = f($j)yl)' (a) Show that fjl is periodic in both indices. (b) Describe briefly the consequences of the periodicity identified in (a) for numerical solution of the Poisson equation using discrete Fourier transforms. (c) Show that the transforms defined by Eqs. (25) and (26) are exact inverses. (d) Consider application of the discrete Fourier transform to the Poisson equation 52¢ (92¢ a? + Bil—2 = "*0" (27) using centred differences with equal grid spacing h in a: and y to approximate the derivatives. Show that a discrete Fourier transform solution is provided by ~ _ 2 gm” — cos — 1)] + cos —1)]— 2' (28) (15 marks) ...
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093983 - 39/83 Semester 2, 2009 Page 1 of 8 THE UNIVERSITY...

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