Numerical Analysis II, Exam End Term Spring 2017
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1 H. Ammari W. Wu S. Yu Spring Term 2017 Numerical Analysis II ETH Zürich D-MATH End Term Spring 2017 Problem 1 Consider dx = f(t, x), t [0, T ] dt x(0) = x 0 R [28 Marks] with f C subject to the Lipschitz condition f(t, x) f(t, y) C x y for x, y R, t [0, T ]. We use the following numerical scheme for (1.1): (1.1) x k+1 = x k tf(t k, x k ) (1.2) (1a) Define the Truncation error by T k ( t) = prove that the scheme (1.2) is of order 2. 透き通る波映る僕らの影は蒼く遠くあの日僕は世界を知りそれは光となった僕は歌うよ笑顔をくれた君が泣いてるときほんのすごしだけでもいい君の支えになりたい僕が泣いてしまった日に君がそうだったように遠く離れた君のもとへこの光が空を越えて羽ばたいてゆくそんな歌をて届けたい透き通る波映る僕らの影は蒼く遠くあの日僕は世界を知りそれは光となった僕は歌うよ x(t + t) x(t t) 2 tf(t, x), 2 t End Term Spring 2017 Page 1
2 (1b) What kind of method (1.2) is? Explicit One-step Implicit One-step Explicit Two-step Implicit Two-step (1c) Is scheme (1.2) consistent with (1.1)? Prove that. 透き通る波映る僕らの影は蒼く遠くあの日僕は世界を知りそれは光となった僕は歌うよ笑顔をくれた君が泣いてるときほんのすごしだけでもいい (1d) Suppose that {x 2,, x k } and { x 2,, x k } have been generated by (1.2) but with different initial data x 0, x 1 and x 0, x 1. Prove by induction that Is (1.2) stable? Prove that. x k x k C max{ x 0 x 0, x 1 x 1,, x k 1 x k 1 }. 透き通る波映る僕らの影は蒼く遠くあの日僕は世界を知りそれは光となった僕は歌うよ笑顔をくれた君が泣いてるときほんのすごしだけでもいい君の支えになりたい僕が泣いてしまった日に君がそうだったように遠く離れた君のもとへこの光が空を越えて羽ばたいてゆくそんな歌をて届けたい君の支えになりたい僕が泣いてしまった日に君がそうだったように遠く離れた君のもとへこの光が空を越えて羽ばたいてゆくそんな歌をて届けたい僕が泣いてしまった日に君がそうだったように End Term Spring 2017 Page 2
3 (1e) Is (1.2) convergent? Prove that. 透き通る波映る僕らの影は蒼く遠くあの日僕は世界を知りそれは光となった僕は歌うよ笑顔をくれた君が泣いてるときほんのすごしだけでもいい君の支えになりたい僕が泣いてしまった日に君がそうだったように遠く離れた君のもとへこの光が空を越えて羽ばたいてゆくそんな歌をて届けたい Problem 2 Consider the scheme where [13 Marks] x k+1 = x k + t 2 (κ 1 + κ 2 ) (2.1) κ 1 = f(t k, x k ), κ 2 = f(t k+1, x k + tκ 1 ). (2a) Is (2.1) a one-step method or two-step method? One-step method Two-step Method (2b) Define the truncation error by T k ( t) = x(t + t) x(t) t 1 [f(t, x(t)) + f(t + t, x(t) + tf(t, x(t)))]. 2 Prove that (2.1) is consistent and of at least order 2 by either direct calculation or properties from Runge-Kutta method. Attention: Please write out coefficients {a ij }, {b i }, {c i } and order conditions explicitly(no need to prove) if choosing to use RK properties. End Term Spring 2017 Page 3
4 透き通る波映る僕らの影は蒼く遠くあの日僕は世界を知りそれは光となった僕は歌うよ笑顔をくれた君が泣いてるときほんのすごしだけでもいい君の支えになりたい僕が泣いてしまった日に君がそうだったように遠く離れた君のもとへこの光が空を越えて羽ばたいてゆくそんな歌をて届けたい透き通る波映る僕らの影は蒼く遠くあの日僕は世界を知りそれは光となった僕は歌うよ笑顔をくれた君が泣いてるときほんのすごしだけでもいい君の支えになりたい僕が泣いてしまった日に君がそうだったように遠く離れた君のもとへこの光が空を越えて羽ばたいてゆくそんな歌をて届けたい透き通る波映る僕らの影は蒼く遠くあの日僕は世界を知りそれは光となった僕は歌うよ笑顔をくれた君が泣いてるときほんのすごしだけでもいい君の支えになりたい僕が泣いてしまった日に君がそうだったように遠く離れた君のもとへこの光が空を越えて羽ばたいてゆくそんな歌をて届けたい透き通る波映る僕らの影は蒼く遠くあの日僕は世界を知りそれは光となった僕は歌うよ End Term Spring 2017 Page 4
5 Problem 3 [19 Mark] Consider the Hamiltonian system dp dt = q, (3.1) dq dt = p, where p(t), q(t) R and the numerical scheme { p k+1 = p k tq k (3.2) q k+1 = q k + tp k+1. (3a) What kind of method (3.2) is? Explicit Euler method Explicit Leapfrog method Implicit Euler method Implicit Leapfrog method (3b) Define the numerical flow Φ t : (p k, q k ) (p k+1, q k+1 ). Prove that (3.2) is symplectic, i.e., the numerical flow satisfies ( ) 0 I (Φ t) T JΦ t = J, where J =. I 0 透き通る波映る僕らの影は蒼く遠くあの日僕は世界を知りそれは光となった僕は歌うよ笑顔をくれた君が泣いてるときほんのすごしだけでもいい君の支えになりたい僕が泣いてしまった日に君がそうだったように遠く離れた君のもとへこの光が空を越えて羽ばたいてゆくそんな歌をて届けたい僕は歌うよ笑顔をくれた君が泣いてるときほんのすごしだけでもいい君の支えになりたい遠く離れた君のもとへこの光が空を越えて羽ばたいてゆくそんな歌をて届けたい僕は歌うよ笑顔をくれた君が泣いてるとき End Term Spring 2017 Page 5
6 (3c) Is the Hamiltonian H associated with (3.1) preserved by the scheme (3.2)? Prove that. 透き通る波映る僕らの影は蒼く遠くあの日僕は世界を知りそれは光となった僕は歌うよ笑顔をくれた君が泣いてるときほんのすごしだけでもいい君の支えになりたい僕が泣いてしまった日に君がそうだったように遠く離れた君のもとへこの光が空を越えて羽ばたいてゆくそんな歌をて届けたい透き通る波映る僕らの影は蒼く遠くあの日僕は世界を知りそれは光となった僕は歌うよ笑顔をくれた君が泣いてるときほんのすごしだけでもいい君の支えになりたい僕が泣いてしまった日に君がそうだったように遠く離れた君のもとへこの光が空を越えて羽ばたいてゆくそんな歌をて届けたい End Term Spring 2017 Page 6
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