Advanced methods for ODEs and DAEs Lecture 9: Multistep methods/ Differential algebraic equations Bojana Rosic, 22. Juni 2016
PART I: MULTISTEP METHODS 22. Juni 2016 Bojana Rosić Advanced methods for ODEs and DAEs Seite 2
Multistep methods Numerical methods for solving ODE x = f (x, t) can be classified as one step methods : these methods use only information from one time point to compute the next xn+1 = xn + hf (tn, xn ) requires nowledge on xn multistep methods: these methods require nowledge on more than one time point 3 1 xn+1 = xn + h fn fn 1 requires nowledge on xn, xn 1 2 2 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 3
Multistep methods (MM) To integrate ODE x = f (x, t) one may use explicit MM : interpolate f (rhs/derivative) by using points xn = x(tn ), xn 1 = x(tn 1 ),..., xn +1 = x(tn +1 ) implicit MM: interpolate f (rhs/derivative) by using points xn+1 = x(tn+1 ), xn = x(tn ),,..., xn +1 = x(tn +1 ) such that P(ti ) = f (ti, x(ti )) and Z tn+1 x(tn+1 ) x(tn j ) = Z tn+1 f (t, x(t))dt = tn j P(t)dt, tn j 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 4 j > 0, j 6
Multistep methods (MM) in which for explicit MM : P(t) = p fn `n = =0 p p Y t tn l. tn tn l l=0 fn =0 l6= p x(tn+1 ) x(tn j ) = wjnp fn =0 and for implicit MM: P(t) = p = 1 fn p Y l= 1 l6= x(tn+1 ) x(tn j ) = t tn l. tn tn l p wjnp fn = 1 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 5
Examples Adams-Bashford formulas (explicit) : =1: =2: =3: =4: xn+1 xn+1 xn+1 xn+1 = xn + hfn, = xn + h 32 fn 12 fn 1, 23 16 = xn + h 12 fn 12 fn 1 + 59 = xn + h 55 f 24 n 24 fn 1 + 5 12 fn 2 37 24 fn 2, 9 24 fn 3 Adams-Moulton formulas (implicit): =1: xn+1 =2: xn+1 =3: xn+1 h (fn+1 + fn ), 2 5 8 1 = xn + h 12 fn+1 + 12 fn 12 fn 1, h (9fn+1 + 19fn 5fn 1 + fn 2 ) = xn + 24 = xn + 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 6
Bacward differentiation formulas Similar to Adams-Moulton formulas with the only difference that the solution is being interpolated, i.e. (tn+1, x(tn+1 )),..., (t p, x(t p )) are used for the approximation of the solution x(t) P(ti ) = x(ti ) and not its derivative f (t, x). However, one also has to satisfy dp (tn+1 ) = f (tn+1, xn+1 ). dt 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 7
Example Interpolate x by a polynomial P = at + b such that 1) P(tn ) = atn + b = xn, 2) P(tn+1 ) = atn+1 + b = xn+1 and d (at + b) = a, dt By susbtructing 1) and 2) we have P (t) = a= x t = f (t, x) xn+1 xn xn+1 xn = P (tn+1 ) = a = f (tn+1, xn+1 ) tn+1 tn h f (tn+1, xn+1 ) = xn+1 xn xn+1 = xn + hf (tn+1, xn+1 ) tn+1 tn and hence one has obtained the first order BDF method. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 8
Bacward differentiation formulas In a similar manner by taing higher number of nown points and polynomial order one may derive the general formula s a xn+ = hβf (tn+s, xn+s ), =0 which further may loo lie s s s =1: =2: =3: xn+1 xn+2 43 xn+1 + 31 xn 9 2 xn+3 18 11 xn+2 + 11 xn+1 11 xn = xn + hf (tn+1, xn+1 ) = 32 hf (tn+2, xn+2 ) 6 = 11 hf (tn+3, xn+3 ) 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 9
Linear multistep methods (LMM) For a system of equations x = f(t, x), x(t0 ) = x0, the general linear multistep method reads i=0 ai xn+i = h bi fn+i, (1) i=0 in which ai and bi are the coefficients of the -step method and fj := f(tj, xj ). On the other side, the exact solution satisfies: x a (tn ) = f(tn, xa (tn )) 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 10
Accuracy and consistency order The local error reads loc = xa (tn+ ) x(tn+ ) in which a x(tn+ ) = 1 j=0 aj xa (tn+j ) + h bj f (tn+j, xa (tn+j )) j=0 By following derivation given in Lecture 12 (ODE1), one finally obtains 2 j loc = aj xa (tn ) + h (jaj bj ) xa0 (tn ) + h2 ( aj jbj ) xa00 (tn ) 2 j=0 j=0 j=0 q+1 q j j (q+1) + + hq+1 [ ( aj bj ) xa (tn ) + O(hq+2 ) (q + 1)! q! j=0 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 11
Accuracy and consistency order The linear multistep is consistent if lim h 0 loc =0 h loc 1 = aj xa (tn ) + (jaj bj ) xa0 (tn ) h h j=0 j=0 j2 +h ( aj jbj ) xa00 (tn ) + + 2 j=0 q+1 q j j (q+1) + + h q [ ( aj bj ) xa (tn ) + O(hq+1 ) (q + 1)! q! j=0 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 12
Accuracy and consistency order Hence, the method is consistent of order p if j=0 and aj = 0, (jaj bj ) = 0 j=0 jq j q 1 ( aj bj ) = 0, (q)! (q 1)! q = 2, p j=0 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 13
Exercise For the method 32 xn 2xn 1 + 12 xn 2 = hfn one has a0 = 1/2, a1 = 2, a2 = 3/2, b0 = b1 = 0, b2 = 1. Hence, al = 0 = a0 + a1 + a2 = 1/2 + 2 3/2 = 0 l=0 (lal bl ) = 0 = a1 + 2a2 b2 = 2 + 3 1 = 0 l=0 1 ν 1 l al l ν 1 bl = 0, ν! (ν 1)! ν=2 l=0 a1 /2 + 4a2 /2 2b2 = 1 + 3 2 = 0 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 14
Zero stability In order to chec if the scheme is convergent, one has to investigate zerostability. This stability is defined on ODE x = 0 After applying LMM, one obtains ρ(ξ) := al xn+l = 0 l=0 which becomes difference equation with general solution: xn = r pi ξni, i=1 where ξi are the roots of the first characteristic polynomial ρ. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 15
Zero stability A scheme satisfies the root condition (is zero stable) if every root ξi of the first characteristic polynomial ρ(ξ) = ai ξi. i=0 has magnitude smaller than one, ξi 6 1, and if every root ξi with ξi = 1 is a simple root of ρ. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 16
Convergence Definition In order that a consistent scheme is convergent, the following stability property has to be fulfilled: A scheme satisfies the root condition (is zero stable) if every root ξi of the first characteristic polynomial ρ has magnitude smaller than one, ξi 6 1, and if every root ξi with ξi = 1 is a simple root of ρ. However, this defines the stability with respect to the initial conditions. As suspected, this is not enough. One also has to consider stability with the respect to the step size and properties of the system being integrated. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 17
Stability Consider the Dahlquist problem x = λx, u(0) = 1 and the linear multistep method ai xn+i = h i=0 It follows i=0 ai xn+i = hλ bi fn+i i=0 bi xn+i i=0 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 18
Stability The last one is the difference equation whose solution is power function xn = cξn. By putting the ansatz into the left side of equation one obtains the first characteristic equation ρ(ξ) := ai ξi i=0 whereas in the right hand side the second characteristic polynomial σ(ξ) := ai ξi i=0 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 19
Stability Taing that z := λh one may rewrite i=0 ai xn+i = hλ bi xn+i i=0 to 0= (ai zbi )ξi = ρ(ξ) zσ(ξ) i=0 Hence, a stable numerical solution must have the roots which satisfy ξl 6 1 If ξl is a root of multiplicity > 1 then ξl < 1 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 20
Stability region The stability region of the linear multistep method i=0 ai xn+i = h bi fn+i i=0 is defined by Gs := {z : C ; ξl 6 1and if ξl is a root of multiplicity > 1 then ξl < 1} 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 21
Root locus curve Note, that on the boundary of Gs at least one root satisfies ξl = 1. Then it follows ρ(ξ) ρ(eiϕ ) z= =, σ(ξ) σ(eiϕ ) where ξ = eiϕ. The root locus curve is defined by C := z C : z= ρ(eiϕ ), ϕ [0, 2π] σ(eiϕ ) 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 22
Stability region 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 23
A-stability A linear multistep method is called A-stable, if all roots of ρ(ξ) zσ(ξ) = 0 are in region defined by Gs := {z : C ; ξl 6 1and if ξl is a root of multiplicity > 1 then ξl < 1} 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 24
Stability region From the previous slide, one may conclude Implicit methods have a larger region of absolute stability than a corresponding explicit method of the same order The absolute stability region typically becomes smaller as order increases BDF methods have the largest stability region compared to other LMM 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 25
Problem Theorem No explicit LMM is A-stable. The order of an A-stable LMM cannot exceed two. The second order A-stable LMM with the smallest error coefficient is the trapezoidal rule. Note that no such restriction exists for implicit Runge-Kutta methods. In order to obtain high orders of accuracy with LMMs, we will have to relax our stability demands and give up a portion of the left-half plane. Proof: see Grigorieff (1977) or Hairer/Wanner 1996. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 26
A(α)-stability A numerical scheme is called A(α)-stable, if R0 (z) 6 1 for all z C with arg(z) π 6 α, α (0, π/2). (see Widlund, 1967) Im z 1111111111111 0000000000000 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 Re z Im z 1111111111111 0000000000000 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 α 0000000000000 1111111111111 α 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 0000000000000 1111111111111 Note, that A(α)-stable with α = π/2 is A-stable. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 27 Re z
A(α)-stability For BDF-methods we get s α 1 90 2 90 3 86.03 4 73.35 5 51.84 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 28 6 17.84
Implementation of BDF methods When applied on a system of ODEs x = f(t, x), BDF methods lead to a non-linear system of the form G(xn+ ) := xn+ h b0 fn+ + rn+ = 0, a0 where rn+ includes the values xn+l, l = 0,..., 1. There are at least two possiblities to solve the system: by fixed point iteration and Newton-lie method. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 29
Implementation of BDF methods However, we show that for the stiff problems the Newton method is the only one that can be applied since a fixed point iteration may lead to numerical (0) instabilities. The starting values xn+ for Newton s method should not be choosen with an explicit predictor scheme, since numerical instabilities may arise. It is better to interpolate the nown values xn+l. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 30
Fixed point iteration Aplying fixed point iteration leads to () xn+1 = () ai xn+i + hb0 f (tn+1, xn+1 ) + i=1 bi fn+i i=1 According to Banach fixed point theorem, this iteration is convergent only when Lh b0 < 1 in which L is the Lipschitz constant. For non-stiff problems L is small and the step size is chosen according to accuracy conditions. However, for stiff problems the step size is very much restricted by this condition. In this case Newton iteration has much better performance. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 31
Comparison of one and multistep methods One step methods: To achieve higher orders with one-step methods additional right hand side evaluations are necessary, which might be very inefficient. Consistency is harder to analyze than stability. The analysis is restricted to only one step, which maes a change of step size easier. Linear multistep methods: With linear multistep methods there is an urgent need for appropriate good starting values. There is an intensive use of former inexact values. A change of step size is rather expensive. Stability is easier to analyze in contrast to consistency. In many cases only one additional right-hand side evaluation is needed in each step. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 32
PART II: DIFFERENTIAL ALGEBRAIC EQUATIONS 22. Juni 2016 Bojana Rosić Advanced methods for ODEs and DAEs Seite 33
What you need to now before this lecture 1. definition of ODEs 2. Lipschitz continuity 3. Inverse theorem 4. Implicit function theorem 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 34
Lipschitz continuity A function f is said to be continuously differentiable if the derivative f 0 (t) exists and is itself a continuous function. Lipschitz continuity is a weaer condition than continuous differentiability. A Lipschitz continuous function is pointwise differentiable almost everwhere and wealy differentiable. The derivative is essentially bounded, but not necessarily continuous. Definition A function f : [a, b] R is uniformly Lipschitz continuous on [a, b] (or Lipschitz, for short) if there is a constant C such that f (x) f (y) 6 C x y for all x, y [a, b]. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 35
Lipschitz continuity Example: a function f = t is not strongly but wealy differentiable. Why? Because the derivative does not exist at t = 0. Namely, the limit t + h t lim h 0 h at t = 0 loos lie 0 + h 0 h = lim h 0 h 0 h h lim and hence, does not exist. But the function is wealy differntiable. It has derivative everywhere except at t = 0. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 36
Inverse function theorem Let F be a C, > 1 map from an open neighborhood of p0 Rn Rn ; with q0 = F (p0 ). Suppose the derivative DF (p0 ) is invertible. Then there is a neighborhood U of p0 and a neighborhood V of q0 such that F : U V is one-to-one (injective) and onto (surjective), i.e. bijective, and F 1 : V U is a C map. (One says F : U V is a diffeomorphism.) 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 37
Inverse function theorem 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 38
Implicit function theorem Suppose U is a neighborhood of x0 Rm ; V a neighborhood of y0 R` ; and we have a C map F : U V R` Assume Dy F (x0 ; y0 ) is invertible. Then the equation F (x; y) = u0 defines y = g(x; u0 ) for x near x0 (satisfying g(x0 ; u0 ) = y0 ) with g a C map. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 39
Implicit function theorem Let us observe function F (x, y) = x 2 + y 2 1 which satisfies F (x, y) = 0. The derivative is Dy F = 2y for which one may distinguish two cases: 1. when y > 0 Dy F > 0 y = p 1 x2 2. and when y < 0 p Dy F < 0 y = 1 x 2 Only when Dy F 6= 0 one may express y as a function of x. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 40
Explicit-implicit ODEs Until now we have studied ordinary differential equations which can be given in explicit x = f (x, t) or implicit F (t, x, x ) = 0 form. Most of time we were focusing on the first group of ODEs. However, in a similar manner one can solve the second one. Namely, one only has to transform the implicit equation to the explicit one. This actually means that one has to solve F (t, x, x ) = 0 for x. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 41
Inverse and implicit function theorem Provided that we have some continuity in F and non-zero denomenator J= F 6= 0, x one may state x = f (x, t) This then allows us to apply the numerical scheme directly on the explicit case. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 42
Exercise: Example I Let be given the implicit formula x1 x 1 + 1 = 0 x 1 x2 + x 2 + 2 = 0 such that 1 x2 0 x 1 1 x1 = 1 x 2 2 holds. Hence, in order to obtain explicit formula x = f(x, t) one would have to invert 1 0 J= x2 1 Because det J 6= 0, this is possible. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 43
Exercise: Example II Let be given the implicit formula x1 x 1 + 1 = 0, x 1 x2 + 2 = 0 such that 1 x2 0 0 1 x1 x 1 = x 2 2 holds. Notice that Jacobian 1 J= x2 0 0 is not invertible any more because det J = 0. Does this means that we have a problem??!! 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 44
Is it really? 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 45
Differential algebraic equations Hence, the problem given by F (x, x, t) = 0 for which J = x F is singular, is not ODE but differential algebraic equation (DAE). ODEs are only special cases of DAE for which Jacobian is regular. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 46
Exercise: Go bac to example II Note that in x1 x 1 + 1 = 0 x 1 x2 + 2 = 0 the Jacobian 1 J= x2 0 0 is singular without any influence of the variable x2. Also, the derivative of x2 does not appear in equations. This means that we do not have two differential equations. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 47
Exercise: Go bac to example II By resolving the first equation x 1 = x1 + 1 and substituting in (x1 + 1)x2 + 2 = 0 one obtains one differential followed by an algebraic equation. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 48
Why studying DAEs DAEs are very often appearing in engineering practice. Some of examples are mutlibody systems (robots, cars, leg prosthesis etc.) chemical industry electrical engineering etc. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 49
Planar pendulum By arranging balance differential equations mx = Fx my = Fy mg Iy α = M Fx` cos α Fy ` sin α together with algebraic constaints x = ` sin α y = ` cos α one obtains differential algebraic equations. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 50
Electrical circuit Ohm s law: C v c = ic, Lv L = il, vr = RiR together with Kirchhoff s voltage law: vr + vl + vc = v(t) maes an DAE system. Here, vr, vl, vc are the voltages across R,L and C respectively and v(t) is the time varying voltage from the source. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 51
Chemical reaction Robertson reaction describes the inetics of an autocatalytic reaction A 1 B B + B 2 C + B B + C 3 A + C A0 = 0.04A + 1 104 BC B 0 = 0.04A 1 104 BC 3 107 B 2 A+B+C =1 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 52
Classification DAEs can be implicit F (x, x, t) = 0 and semi-explicit ones x = f (t, x, y) 0 = g(t, x, y) in which x is dependent/differentiable variable, y is independent/algebraic variable, and g(t, x, y) is the algebraic constraint. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 53
Classification DAEs can be broadly classified to nonlinear F (x, x, t) = 0 F is nonlinear w.r.t. any of arguments and linear ones A(t)x + B(t)x(t) = f (t) 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 54
Classification Furthermore, DAEs can be classified according to how far are they from pure ODEs or pure algebraic equations. Thus, the equations can be classified according to differentiation index perturbation index Kronecer index tractability index geometric index strangeness index 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 55
Index Solving a DAE is more difficult than solving ODE. Namely, the solution depends on derivatives of the model equations or input functions. Also, the algebraic equations restrict the dynamics of the system and sometimes it is difficult to fullfil them numerically. Initial conditions are another problem because they need to agree with the algebraic constraints at each time moment. All of these and some more difficulities are measured via previously mentioned incdices. Hence, indices can be udnerstood as some ind of measure of difficulty in solving DAE. The higher index is, the more difficult is to solve the corresponding DAE. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 56
Differentiation index Definition The nonlinear DAE F (x, x, t) = 0 has differentiation index id if id is the minimal number of differntiations df = 0, dt d 2F = 0, dt 2 d id F =0 dt id such that equations allow to extract an explicit ordinary differential system x = f (x, t) using only algebraic manipulations. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 57
Differentiation index it counts the minimal number of times that all or part of the equations in the system must be differentiated in order to obtain pure ODE. Hence, it measures how far the given DAE system is from an equivalent explicit ODE system. Higher differentiation index more difficult to solve DAE. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 58
Exercise Determine differentiation index of following DAE u = cos(t) u(t) z (t) = 0 Answer: Writing equations in the following form u = cos(t) z = u(t) one may notice that id = 0. Hence, this is ODE. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 59
Exercise Determine differentiation index of following DAE u=t Answer: After differentiation of previous equation, one obtains ODE u = 1. Hence, id is 1. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 60
Exercise Determine differentiation index of following DAE u1 = t 2 u2 = u 1 Answer: After differentiation of the first equation one obtains u 1 = 2t u2 = 2t and similarly u 1 = 2t u 2 = u 1 = 2. Hence, id is 2. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 61
Exercise Determine differentiation index of following DAE u 1 = u3 0 = u2 (1 u2 ) 0 = u1 u2 + u3 (1 u2 ) t From second equation one may notice that u2 can be either 0 or 1. Hence, the differentitaion index will depend on u2 value: I case: u2 = 0 u3 = t and u 1 = u3. Hence, after one differentiation (id = 1) one obtains u 1 = t and u 3 = 1. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 62
Exercise II case u2 = 1 u1 = t and u 1 = u3. Hence, after differentiating u1 = t one obtains u 1 = 1 and hence 1 = u3. After differentiation of 1 = u3 one obtains u 3 = 0 and u 1 = 1. Thus, id = 2. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 63
Maximal differentation index Let F be a family of right-hand sides such that, for any δ F, the DAE has only one solution. Then the maximum differentiation index is =δ. imd := max id is the differentiation index of F (t, x, x ) δ F 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 64
Perturbation index This index is a measure of sensitivity of the solutions with respect to the perturbations of the given problem. Before we give definition of the perturbation index let us observe the semi-explicit system y 0 = f (y, z) 0 = g(y, z) and perturb it such that y 0 = f (y, z ) + δ1 (t) 0 = g(y, z ) + δ2 (t) holds. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 65
Perturbation index If gz is invertible then one may express variables z and z (Implicit function theorem) from 0 = g(y, z) and 0 = g(y, z ) + δ2 (t). Using Lipschitz continuity condition this further lead us to z z 6 C1 (y y + δ2 (t)) 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 66
Perturbation index Further one may substract y 0 = f (y, z) from y 0 = f (y, z ) + δ1 (t) such that y 0 y 0 = f (y, z ) + δ1 (t) f (y, z) holds. After integration of the last equation from 0 to t, one obtains Zt Zt Zt y y = y (0) y(0) + f (y, z )dt + δ1 (t)dt f (y, z)dt 0 0 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 67 0
Perturbation index Lipschitz condition for Zt Zt f (y, z )dt f (y, z)dt 0 0 reads Zt Zt f (y, z )dt f (y, z)dt 6 C2 0 Zt y ydt + C3 0 z zdt 0 where we have already derived that z z 6 C1 (y y + δ2 (t)) Hence, Zt Zt f (y, z )dt f (y, z)dt 6 C4 0 Zt y ydt + C3 0 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 68 δ2 (t)dt 0
Perturbation index Hence, from Zt y y = y (0) y(0) + Zt f (y, z )dt + 0 Zt f (y, z)dt δ1 (t)dt 0 0 follows Zt y y 6 y (0) y(0) + Zt f (y, z ) f (y, z)dt + 0 δ1 (t)dt 0 i.e. y ydt + C3 y y 6 y (0) y(0) + C4 Zt Zt Zt δ2 (t)dt + 0 0 6 C5 (y (0) y(0) + supδ2 + supδ1 ) τ I τ I 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 69 δ1 (t)dt 0
Perturbation index-definition Starting from the DAE F (x, x, t) = 0, we introduce a perturbation δ F of the right-hand side. Loo for a solution x of the DAE x, t) = δ, F (x, in which F is a family of right-hand sides such that for any δ the perturbed DAE has only one solution. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 70
Perturbation index-definition Definition The DAE has the perturbation index ip along the solution x on [t0, t], if ip is the smallest integer such that for all x having a defect δ there exists an estimate of the form x (t) x(t) Zt 6 C x (0) x(0) + sup τ I δ(τ) dτ + t0 ip j=1 j 1 δ(τ). sup ( τ)j 1 τ J on [0, T ]. Here, C is a constant that depends on F, the length I of the time interval [t0, t], and on the solution x. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 71
Remars For some nonlinear DAEs, the class of perturbations F may have only few elements. It may be possible that such an estimate does not exist. Often only perturbations with δ 6 1 are considered. This restriction can be motivated by the numerical bacground. One is interested in the behavior of a numerical method if the right hand side is disturbed with small perturbations. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 72
Perturbation index Going bac to our example: y y 6 C5 (y (0) y(0) + supδ2 + supδ1 ) τ I one concludes that the perturbation index is ip = 1. 22. Juni 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 73 τ I