NUMERICAL ANALYSIS Chapter 5 Ordinary Differential Equation http

NUMERICAL ANALYSIS Chapter 5. Ordinary Differential Equation 수학과 김찬용 , 컴퓨터학과 김현우, 장한용 http: //korea. ac. kr CSE 503 응용 수학Ⅱ

5. 1 The Elementary Theory of Initial. Value Problems n Definition 5. 1 n f(x, y) : Lipschitz condition on set D⊂R 2 , ∃L > 0 with , (t, y 1), (t, y 2) ∈ D , L : Lipschitz constant n Definition 5. 2 n D⊂R 2 : convex , (t 1, y 1), (t 2, y 2) ∈ D , λ ∈[0, 1] ( (1 - λ)t 1 + λt 2 , (1 - λ) y 1 + λy 2 )∈ D i. e. D = { (t , y) | a ≤ t ≤ b, | y | ＜ ∞ } : convex http: //korea. ac. kr CSE 503 응용 수학Ⅱ 2

5. 1 The Elementary Theory of Initial. Value Problems n Definition 5. 3 n f(x, y) is defined on a convex set D⊂R 2 ∃L > 0 with => f : Lipschitz condition on D with Lipschitz constant L. n Definition 5. 4 , f(x, y) : continuous on D n If f satisfies a Lipschitz condition on D, then y′(t) = f(t, y) , a ≤ t ≤ b, y(a) =a has a unique solution y(t) for a ≤ t ≤ b. http: //korea. ac. kr CSE 503 응용 수학Ⅱ 3

5. 1 The Elementary Theory of Initial. Value Problems n Definition 5. 5 : well-posed problem n if ∃y(t) : unique solution, and ∃ε 0 > 0 , ∃k > 0 s. t ∀ε, with ε 0 > ε > 0, whenever δ(t) : continuous with |δ(t)| < ε for all t in [a , b] & when |δ 0| < ε, dz/dt = z′(t) = f(t, z) + δ(t), a ≤ t ≤ b, z(a) = a + δ 0 has unique solution z(t) s. t |z(t) - y(t)| < kε for all t in [a , b] n Definition 5. 6 n b = { (t, y) | a ≤ t ≤ b, |y| < ∞ } f : continuous & Lipschitz condition => dy/dt = f(t, y) , a ≤ t ≤ b, y(a) = a : well-posed http: //korea. ac. kr CSE 503 응용 수학Ⅱ 4

5. 2 Euler’s Method n dy/dy = y′(t) = f(t, y) , a ≤ t ≤ b , y(a) = a ti∈[a, b] : mesh points. ti = a + ih , for each i = 0, 1, 2, … , N ( h= (b-a)/N = ti+1 – ti : step size) using Taylor’s Theorem, y(t) ∈ C 2[a, b] : unique solution, ∈[a, b] since h= ti+1 – ti n http: //korea. ac. kr Euler’s method : CSE 503 응용 수학Ⅱ 5

5. 2 Euler’s Method n Lemma 5. 7 n n ≥ -1 & ∀x > 0, 0≤ (1+x)m ≤ emx Lemma 5. 8 n n ∀x s, t ∈ R , then : Theorem 5. 9 n f : continuous & Lipschitz condition with L on D & ∃M with |y˝(t)| ≤ M, for all t∈[a, b]. Let y(t) : unique solution, Euler’s method => http: //korea. ac. kr CSE 503 응용 수학Ⅱ 6

5. 2 Euler’s Method n Theorem 5. 10 n let y(t) : unique solution & u 0, u 1, … , un : approximation, & |y˝(t)| ≤ M then n n http: //korea. ac. kr : minimal value of E(h) CSE 503 응용 수학Ⅱ 7

5. 2 High-Order Taylor Methods n Definition 5. 11 has local truncation error for each i = 0, 1, … , N -1 Taylor method of order n ω0 = a , ωi = ωi + h. T(n)(ti, ωi), for each i = 0, 1, … , N -1 where T(n)(ti, ωi) = f(ti, ωi) + h/2*f ′(ti, ωi) + … + hn-1/n!*f(n-1)(ti, ωi) Note : Euler’s method is Taylor’s method of order one. http: //korea. ac. kr CSE 503 응용 수학Ⅱ 8

5. 2 High-Order Taylor Methods n Definition 5. 12 n using Taylor’s method’s of order n, h: step size. if y ∈ Cn+1[a, b], then the local truncation error is O(hn). http: //korea. ac. kr CSE 503 응용 수학Ⅱ 9

5. 4 Runge-Kutta Methods n Definition 5. 13 n f(t, y) & all its partial derivatives of order less than or equal to n+1 : continuous on let , ∀ , ∃ ∈(t, t 0), ∃ ∈(y, y 0) with where : n th Taylor polynomial in two variables. http: //korea. ac. kr CSE 503 응용 수학Ⅱ 10

5. 4 Runge-Kutta Methods ≒ a 1 T(2)(t, y) + a 1 b 1(t+a 1, y+b 1) n where http: //korea. ac. kr CSE 503 응용 수학Ⅱ 11

5. 4 Runge-Kutta Methods => where http: //korea. ac. kr CSE 503 응용 수학Ⅱ 12

5. 4 Runge-Kutta Methods n Specific Runge-Kutta method. n Midpoint Method n Modified Euler Method n Heun’s Method http: //korea. ac. kr CSE 503 응용 수학Ⅱ 13

5. 4 Runge-Kutta Methods n Runge-Kutta Order Four : for each i = 0, 1, … , N-1 http: //korea. ac. kr CSE 503 응용 수학Ⅱ 14

5. 5 Error Control and the Runge-Kutta. Fehlberg Method n (n+1)st – order Taylor method of the form n Producing approximations assume http: //korea. ac. kr CSE 503 응용 수학Ⅱ 15

5. 5 Error Control and the Runge-Kutta. Fehlberg Method ≒ ≒ http: //korea. ac. kr CSE 503 응용 수학Ⅱ 16

5. 5 Error Control and the Runge-Kutta. Fehlberg Method n Using runge-kutta method with local truncation error of order five, estimate the local error in a runge-kutta method of order four where the coefficient equation are http: //korea. ac. kr CSE 503 응용 수학Ⅱ 17

5. 5 Error Control and the Runge-Kutta. Fehlberg Method The value of q determined at the i th step is used for two purpose n When q<1, to reject the initial choice of h at the i th step and repeat the calculations using qh, and n When q≥ 1, to accept the computed value at the i th step using the step size h and to change the step size to qh for (i + 1)st step. n n=4 runge-kutta-fehlberg method n http: //korea. ac. kr CSE 503 응용 수학Ⅱ 18

5. 6 Multistep Methods n Definition 5. 14 n m-step multistep method are consistants. when bm = 0 : explicit or open bm ≠ 0 : implicit or closed http: //korea. ac. kr CSE 503 응용 수학Ⅱ 19

5. 6 Multistep Methods To begin the derivation of multistep method, Since we can not integrate f(t, y(t)) without knowing y(t) P(t) : interpolating polynomial, (t 0, ω0) …. . (ti, ωi) assume http: //korea. ac. kr CSE 503 응용 수학Ⅱ 20

5. 6 Multistep Methods n Adams-Bashforth explicit m-step technique Pm-1(t) : backward-difference polynomial, …. . t = ti + sh , dt = hds , error term http: //korea. ac. kr CSE 503 응용 수학Ⅱ 21

5. 6 Multistep Methods n Example three-step Adams-Bashforth technique http: //korea. ac. kr CSE 503 응용 수학Ⅱ 22

5. 6 Multistep Methods y i ≒ ωi http: //korea. ac. kr CSE 503 응용 수학Ⅱ 23

5. 6 Multistep Methods n Definition 5. 15 is the (i+1)st step in a multistep method, local truncation error at this step is http: //korea. ac. kr CSE 503 응용 수학Ⅱ 24

5. 6 Multistep Methods Two-step Three-step Four-step http: //korea. ac. kr Adams-Bashforth Adams-Moulton τi+1(h) = 5/12 y(3)(μi)h 2 μi ∈ (ti-1, ti+1) τi+1(h) = -1/24 y(4)(μi)h 3 μi ∈ (ti-1, ti+1) τi+1(h) = 3/8 y(4)(μi)h 3 μi ∈ (ti-2, ti+1) τi+1(h) = -19/720 y(5)(μi)h 4 μi ∈ (ti-2, ti+1) τi+1(h) = 251/720 y(5)(μi)h 4 μi ∈ (ti-3, ti+1) τi+1(h) = -3/160 y(6)(μi)h 5 μi ∈ (ti-3, ti+1) CSE 503 응용 수학Ⅱ 25

5. 7 Variable step-size Multistep Method n Adams-Bashforth four-step method ω0, ω1, . . . , ωi , μi ∈ (ti-3, ti+1) n Adams-Bashforth three-step method http: //korea. ac. kr CSE 503 응용 수학Ⅱ 26

5. 7 Variable step-size Multistep Method new step size qh, generating new approximations As a consequence, we commonly ignore the step-size change when the local truncation error is between ε/10 and ε that is when http: //korea. ac. kr CSE 503 응용 수학Ⅱ 27

5. 8 Extrapolation Methods assume fixed step size h, y(ti)=y(a+h) let h 0 = h/2, use Euler's method with ω0=a , y(a + h 0) = y(a+h/2) apply Midpoint method let h = h/4 use Euler's method ω0=a , y(a + h 1) = y(a+h/4) with ω1, y(a + 2 h 1) = y(a+h/2) with ω2 , y(a + 3 h 1) = y(a+3 h/4) with ω3 http: //korea. ac. kr CSE 503 응용 수학Ⅱ 28

5. 8 Extrapolation Methods approximation http: //korea. ac. kr CSE 503 응용 수학Ⅱ 29

5. 9 High-Order Equations and of Differential Equations n m th - order system http: //korea. ac. kr CSE 503 응용 수학Ⅱ 30

5. 9 High-Order Equations and of Differential Equations n Definition 5. 16 , on satisfy a Lipschitz condition on D, ∃L > 0 with n Definition 5. 17 fi(t, u 1, . . . , um) : continuous on D & satisfy a Lipschitz condition. The system of first-order differential equations, subject to the initial conditions has a unique solution u 1(t), . . . , um(t) for a ≤ t ≤ b. http: //korea. ac. kr CSE 503 응용 수학Ⅱ 31

5. 9 High-Order Equations and of Differential Equations for each i = 1, 2, . . . , m : and then http: //korea. ac. kr CSE 503 응용 수학Ⅱ 32

5. 10 Stability n Definition 5. 18 n n A one-step difference-equation method with local truncation error τi(h) at the i th step is said to be consistent with the differential equation it approximates if Definition 5. 19 n A one-step difference-equation method is said to be convergent with respect to the differential equation it approximates if where yi = y(ti) : exact value of solution of differential equation ωi : approximation obtained from difference method at the with step. http: //korea. ac. kr CSE 503 응용 수학Ⅱ 33

5. 10 Stability n Theorem 5. 20 is approximated by a one-step difference method in the form ∃h 0 > 0, φ(t, w, h) : continuous & satisfies a Lipschitz condition on Then ⅰ) The method is stable; ⅱ) The difference method is convergent if and only if it is consistent, which is equivalent to ⅲ) ∃function and http: //korea. ac. kr CSE 503 응용 수학Ⅱ 34

5. 10 Stability n Theorem 5. 21 with local truncation error τi+1(h) with local truncation error f(t, y) and fy(t, y) : conditinuous on , fy is bounded then, the local truncation error corrector method is http: //korea. ac. kr CSE 503 응용 수학Ⅱ of the predictor- 35

5. 10 Stability n Definition 5. 22 n let λ 1, λ 2, . . . , λm : root of the characteristic equation associated with the multistep difference method & if |λi| ≤ 1 , i = 1, 2, . . . , m, & all roots with absolute value 1 are simple roots, then the difference method is said to satisfy the root condition. http: //korea. ac. kr CSE 503 응용 수학Ⅱ 36

5. 10 Stability n Definition 5. 23 ⅰ) Methods the satisfy the root condition and hand λ = 1 as the only root of the characteristic equation of magnitude one are called strongly stable. ⅱ) Methods the satisfy the root condition and have more than one distinct root with magnitude one are called weakly stable. ⅲ) Methods that do not satisfy the root condition are called instable. n Theorem 5. 24 n A multistep method of the form where : stable ⇔ root condition moreover, if the difference method is consistant with the difference equation, then the method is stable if it is convergent. http: //korea. ac. kr CSE 503 응용 수학Ⅱ 37

5. 11 Stiff Differential Equations n : solution, which the transient solution Euler's method applied to the test equation let h = (b-a) / N , tj = jh , for j = 0, 1, 2, . . . , N , . so absolute error λ < 0 : (ehλ)j decays to zero as j increases |1+hλ| < 1 : proerty approximation => -2 < hλ < 0 This effectively restricts the step size h for Euler's' method to satisfy h < 2/| λ | http: //korea. ac. kr CSE 503 응용 수학Ⅱ 38

5. 11 Stiff Differential Equations ω0 = a + δ 0 : round-off error δ 1 : (1+hλ)jδ 0 : j th step the round-off error since λ < 0 , the condition for the control of the growth of round-off error is the same as the condition for controlling the absolute error, |1+hλ| < 1, which implies that h < 2/| λ | n Definition 5. 24 n The region R of absolute stability for a one-step method is R = { hλ∈C | |Q(hy)| < 1}, and for a multistep method, it is R = { hλ∈C | |bk| < 1, for all zeros bk of Q(z, hy)}. http: //korea. ac. kr CSE 503 응용 수학Ⅱ 39