Advanced methods for ODEs and DAEs

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1 Lecture : Implicit Runge Kutta method Bojana Roić, 9. April 7

2 What you need to know before thi lecture numerical integration: Lecture from ODE iterative olver: Lecture 5-8 from ODE 9. April 7 Bojana Roić Seite

3 Why not explicit Runge Kutta method? Explicit Runge Kutta method can be uccefully ued for integration of non-tiff problem x = x fail to capture the tiff phenomena x = 6 x even when the tep ize i decreaed dratically. (Plauibilize (Sketch) with RHS a vector field and Euler-FW and BW-Method!) 9. April 7 Bojana Roić Seite

4 General implicit Runge Kutta method Runge Kutta method i preented by Butcher table: c a a c a a c a a b b a a..., a b or c A bt Here, A may be uch that aij =, j > i explicit Runge Kutta method (ERK) uch that aij =, j > i and at leat one aii 6= diagonal implicit Runge Kutta method (DIRK) uch that aij =, j > i and all aii 6= and identical ingly diagonal implicit Runge Kutta method (SDIRK) full matrix fully implicit Runge Kutta method (IRK) 9. April 7 Bojana Roić Seite 4

5 Derivation Given firt order differential equation x = f (t, x) the olution hall be numerically evaluated dx = f (t, x) x(tn ) = x(tn )+ dt Z tn f (τ, x)dτ = x(tn )+h tn uch that the local truncation error atifie maxk loc h k 6 Chp 9. April 7 Bojana Roić Seite 5 i= bi k i

Implicit Euler method The implet poible cae i when = and b =, k = f (tn + h, x(tn + h)) i.e., the area under the graph f (τ, x) from tn up to tn i approximated by a right rectangle Z tn f (τ, x)dτ = hf (tn + h, x(tn + h)) = hf (tn, x(tn )) tn uch that xn = xn + hf (tn, xn ).

6 Implicit Euler method The implet poible cae i when = and b =, k = f (tn + h, x(tn + h)) i.e., the area under the graph f (τ, x) from tn up to tn i approximated by a right rectangle Z tn f (τ, x)dτ = hf (tn + h, x(tn + h)) = hf (tn, x(tn )) tn uch that xn = xn + hf (tn, xn ). Thu, the local error i loc = xa (tn ) x(tn ) = xa (tn ) xa (tn ) in which xa (tn ) = xa (tn ) + dxa dt tn (tn 9. April 7 Bojana Roić Seite 6 dxa (tn tn ) = O(h ) dt tn tn ) + O(h )

7 Implicit Euler method 9. April 7 Bojana Roić Seite 7

8 Generaliation: collocation baed RKM To numerically evaluate the integral in Z tn x(tn ) = x(tn ) + f (τ, x)dτ, tn let u define the collocation polynomial P(t) of degree N whoe derivative dp dt coincide with the function f at the ditinct collocation point (tn + ci h) for i =,..., and c,..., c being ditinct real number between and. That i, dp (tn + ci h) = f (tn + ci h, P(tn + ci h)) =: ki, dt and P(tn ) = x(tn ), P(tn + h) = x(tn ). 9. April 7 Bojana Roić Seite 8 i =,...,

9 Generaliation: collocation baed RKM Note that dp dt i the polynomial of mot degree characteried by the mot of coefficient. Thee coefficient can be uniquely determined from given point. In Lecture, ODE, we have hown that the polynomial can be written dp dp (tn +τh) = (tn +ci h)`i (τ) = ki `i (τ) dt dt i= i= in term of Lagrange polynomial `j (τ) = Y i=,i6=j τ ci, cj ci τ [, ] with property `i (cj ) = δij in which δij i the Kronecker delta ymbol. 9. April 7 Bojana Roić Seite 9

10 Generaliation: collocation baed RKM One may write that lope at t + hci i that of P, hence dp (t + cj h) = ki `i (cj ) dt kj = and i= Z tn Z x(tn ) = x(tn ) + f (τ, x)dτ = x(tn ) + h tn dp (tn + τh)dτ dt Integrating Z dp (t + τh)dτ = dt Z ki `i (τ)dτ = i= i= Z ki `i (τ)dτ = i= Z bi = `i (τ)dτ, ki = f (tn + ci h, P(tn + ci h)) 9. April 7 Bojana Roić Seite ki bi

11 Generaliation: collocation baed RKM Furthermore, P(tn + ci h) i unknown in ki = f (tn + ci h, P(tn + ci h)) and can be numerically etimated Z ci P(tn + ci h) = P(tn ) + h = x(tn ) + h Z ci dp (tn + τh)dτ dt kj `j (τ)dτ j= = x(tn ) + h j= Z ci kj `j (τ)dτ = x(tn ) + h j= Z ci aij := `j (τ)dτ aij are the RK coeff! 9. April 7 Bojana Roić Seite kj aij

12 Generaliation: collocation baed RKM Altogether: ki = f (tn + ci h, xn + h i =,..., kj aij ), j= x(tn ) = x(tn ) + h bi ki i= Z ci in which aij = `j (τ)dτ Z bi = `i (τ)dτ Thu, for collocation baed RKM one only need to know ci (quadrature rule). 9. April 7 Bojana Roić Seite

13 Propertie of collocation baed RKM Interpolation of time polynomial up to degree hall be error free τq = cjq `j (τ), q =,..., Hence, q =, Z ci τ dτ = ci = Rci j= τ dτ = ci, and alo Z ci j= P j= aij : `j (τ)dτ = Z ci cj `j (τ)dτ = j= invariance to autonomiation! 9. April 7 Bojana Roić Seite Z ci j= `j (τ)dτ = j= aij,

14 Propertie of collocation baed RKM Similarly, one may how aij cjm = j= a well a cim, m i =,...,, m =,... bj cjm i= = Z i= i= Z = ( `j (τ)dτ)cjm i= Z cjm `j (τ)dτ bi cim aij = bj ( cjm ) m 9. April 7 Bojana Roić Seite 4, τm dτ = = m j =,...,, m =,...,

15 Order condition Thee three equalitie can be gathered to B(p) : bi cim =, m aij cjm = cim, m i= C (q) : j= D(r ) : i= bi cim aij = bj k =,..., p b T c m = m i =,...,, m =,...q ( cjm ) m, j =,...,, m =,..., r with r=q = and p =. D commonly derived by comparion with Taylor. 9. April 7 Bojana Roić Seite 5

16 Order condition The condition B(p) i equivalent to a quadrature rule with the node ci and the weight bi which integrate polynomial of degree p exact. The condition C (q) have the following meaning: the intermediate value ki are integrated exactly by a quadrature rule with the weight aij and the node ci. Thi rule integrate polynomial of degree q exactly. The condition D(r ) mean that the olution x i integrated exactly up to the polynomial degree r. 9. April 7 Bojana Roić Seite 6

17 Order condition Theorem An RK-method with internal tage ha the convergence order p, if the implifying condition B(p), C (l), and D(m) with p 6 min{l + m +, l + } are atified. 9. April 7 Bojana Roić Seite 7

18 Lobatto-IIIA method 9. April 7 Bojana Roić Seite

19 Lobatto-IIIB method 9. April 7 Bojana Roić Seite

20 Lobatto-IIIC method 9. April 7 Bojana Roić Seite

21 General RKM A natural generalization of collocation method i obtained by allowing the cofficient ci,bi, and aij, i =,..., to take on arbitrary value, not necearily related to quadrature rule. In fact, one no longer aume ci to be ditinct. The reult i the cla of general Runge-Kutta method. However, thi tory i not currently the cope of thi ubject. 9. April 7 Bojana Roić Seite

22 Implementation Let u implement RKM on the imple general nonlinear ODE x = f (x, t), x() = x Hence, one obtain kj = f (tn +cj h, xn +h ki aij ), i= x(tn ) = x(tn ) + h bi k i i= 9. April 7 Bojana Roić Seite j =,...,

23 Implementation Implicit Runge Kutta method have nice propertie. However, the main drawback i that one obtain the nonlinear ytem of equation k = f (tn + c h, xn + ha k + + ha k )... k = f (tn + c h, xn + ha k + + ha k ) The dimenion of thi ytem i. 9. April 7 Bojana Roić Seite

24 Implementation In pecial cae, for example in the cae of diagonally implicit Runge Kutta method (A i lower triangular including diagonal), thi ytem plit into non-linear decoupled equation: k = f (tn + c h, xn + ha k )... k = f (tn + c h, xn + ha k + + ha k ) 9. April 7 Bojana Roić Seite 4

25 Implementation To olve thee ytem one require nonlinear iterative olver: fixedpoint iteration (Lecture 5 and 6 from lat emeter) Newton or quai-newton method (Lecture 8) 9. April 7 Bojana Roić Seite 5

26 Fixed point iteration To compute unknown k,..., k one may ue fixed point iteration (ν+) (ν) (ν) k = f tn + c h, xn + ha k + + ha k =: f (k,..., k )... (ν+) k (ν) (ν) = f tn + c h, xn + ha k + + ha k =: f (k,..., k ) () () tarting from k,..., k. 9. April 7 Bojana Roić Seite 6

27 Fixed point iteration Gathering all ki into a vector k = (k,..., k )T, one may rewrite previou equation in a matrix form a f (k,..., k ) f (k,..., k ) k(ν+) = F (k(ν) ) =... f (k,..., k ) 9. April 7 Bojana Roić Seite 7

28 Banach fixed point theorem Theorem Let V be a Banach pace, let I V be a cloed ubet, and let F : I I be a contraction ( 6 q < ) uch that kf (k) F (w)k 6 qkk wk for all k, w I. hold. Then the following concluion hold:.) F ha a unique fixed point k I, i. e. F (k ) = k..) For any k I the equence xn+ = F (xn ) converge to x. Furthermore for differentiable function we aid that (Lecture 6, ODE) q := up kf (ξ)k, ξ = w + η(k w ), η [,] 9. April 7 Bojana Roić Seite 8 w 6ξ6k

29 Convergence Hence, in our cae f / k..... F =. f / k... f / k a fx.... = h.. f / k a fx a fx... a fx in which f (tn + ci h, xn + hai k + + haij kj + + hai k ) fi = = haij fx kj kj Thu, for convergence mut hold q := up kf k < 9. April 7 Bojana Roić Seite 9

30 Exercie Let u integrate ODE x = f (t, x) = te x, x = x(.) =, h = by uing Radau IA method: /4 /4 / /4 5/ /4 /4 i.e. k = f (tn + h, xn + h k h k ) = tn e xn +h 4 k h 4 k 4 4 k = f (tn + h, xn + h k + h k ) = (tn + h)e xn +h 4 k +h 4 k April 7 Bojana Roić Seite

31 Exercie and xn+ = xn + h bi ki = xn + h( k + k ) 4 4 In the firt tep n = one ha k = t e x +h 4 k h 4 k =.e + 4 k 4 k k = (t + h)e x +h 4 k +h 4 k = (.7667)e + 4 k + 4 k Thi i nonlinear ytem of equation which can be olved by fixed point iteration (ν) (ν) (ν+) k =.e + 4 k 4 k (ν+) k (ν) = (.7667)e + 4 k () in which the initial etimate k () = k 9. April 7 Bojana Roić Seite (ν) + 4 k = f (t, x ) =.e.

32 Exercie Thu, the firt fixedpoint iteration i () k () k () =.e + 4 k () 4 k () = (.7667)e + 4 k =.78 () + 4 k =.75 The econd i () k () k () 4 k () =.e + 4 k () = (.7667)e + 4 k =.47 () + 4 k = and o on until m th iteration in which (m+) fix = ki (m) ki (m) ki 9. April 7 Bojana Roić Seite 6 tol, i =,

33 Exercie (m) Once the converged k the firt tep value (m) and k are evaluated, one may etimate (m) (m) bi ki = x + h( k + k ) 4 4 Now the whole proce ha to be repeated for the econd time tep: x = x + h bi ki = x + h( k + k 4 4 in which again k and k are unknown. Hence, again one applie fixed point iteration (ν) (ν) (ν+) k =.e + 4 k 4 k x = x + h (ν+) k (ν) = (.7667)e + 4 k () in which the initial etimate k () = k 9. April 7 Bojana Roić Seite (ν) + 4 k = f (t, x ).

34 Newton method To olve nonlinear equation: k = f (tn + c h, xn + ha k + + ha k )... k = f (tn + c h, xn + ha k + + ha k ) by Newton method, let u rewrite it a f (k ) := k f (tn + c h, xn + ha k + + ha k ) =... f (k ) := k f (tn + c h, xn + ha k + + ha k ) = 9. April 7 Bojana Roić Seite 4

35 Newton method Hence, f (k,..., k ) f (k,..., k ) = G (k) =... f (k,..., k ) The olution i obtained a k(i+) = k(i) (Gk ) (k(i) )G (k(i) ) in which the Jacobian i obtained a f / k.. Gk (k) =. f / k f / k ha fx.... =.. f / k ha fx 9. April 7 Bojana Roić Seite ha fx... ha fx

36 Convergence To invetigate the convergence one may oberve the mapping k(i+) = Φ(k(i) ) = k(i) (Gk (k(i) )) G (k(i) ) and apply fixed point theorem q := up kφ k < See lide from Lecture 8, ODE. 9. April 7 Bojana Roić Seite 6

37 Exercie Let u integrate ODE x = f (t, x) = te x, x = x(.) =, h = by uing Radau IA method: /4 /4 / /4 5/ /4 /4 i.e. k = f (tn + h, xn + h k h k ) = tn e xn +h 4 k h 4 k 4 4 k = f (tn + h, xn + h k + h k ) = (tn + h)e xn +h 4 k +h 4 k April 7 Bojana Roić Seite 7

38 Exercie and xn+ = xn + h bi ki = xn + h( k + k ) 4 4 In the firt tep n = one ha k = t e x +h 4 k h 4 k =.e + 4 k 4 k k = (t + h)e x +h 4 k +h 4 k = (.7667)e + 4 k + 4 k Thi i nonlinear ytem of equation which can be rewritten a: f (k, k ) = k.e + 4 k 4 k = f (k, k ) = k (.7667)e + 4 k + 4 k = and olved by Newton method. 9. April 7 Bojana Roić Seite 8

39 Exercie i.e. (i+) k (i+) k! (i) = k (i) k! (G (k(i) ) with G= = (i) f / k f / k () Starting value are again k (i) () 9. April 7 Bojana Roić Seite 9 (i) = k (i) + 4 k 4 k ) (.e k + 4 k + 4 k ) ((.7667)e k = f (t, x ).! 4 k k (.7667)e + 4 k f / k f / k (.e + 4 k 4 k ) k k +k ) ((.7667)e k (i) k.e + 4 k (i) + 4 k

40 Exercie (m) (m) Once the Newton method ha converged to value k and k, one may etimate the firt tep value (m) (m) x = x + h bi ki = x + h( k + k ) 4 4 Now the whole proce ha to be repeated for the econd time tep: x = x + h bi ki = x + h( k + k 4 4 in which again k and k are unknown. Hence, again one applie Newton method to olve k = t e x +h 4 k h 4 k k = (t + h)e x +h 4 k +h 4 k 9. April 7 Bojana Roić Seite 4

41 Simplified Newton method In Newton method one ha to etimate Jacobian, which i more than expenive. In order to avoid thi, one may ue implified Newton method which etimate Jacobian only at the old time tep The olution i obtained a k(i+) = k(i) (J) G (k(i) ) in which the Jacobian i obtained a ha fx (tn, xn ).. J=. ha fx (tn, xn ) ha fx (tn, xn ).... ha fx (tn, xn ) compute Jacobian firt time and keep it contant for each Newton iteration 9. April 7 Bojana Roić Seite 4

Advanced methods for ODEs and DAEs

Advanced methods for ODEs and DAEs Lecture 13: Overview Bojana Rosic, 13. Juli 2016 Dynamical system is a system that evolves over time possibly under external excitations: river flow, car, etc. The dynamics