Advanced methods for ODEs and DAEs

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1 Advanced methods for ODEs and DAEs Lecture 13: Overview Bojana Rosic, 13. Juli 2016

2 Dynamical system is a system that evolves over time possibly under external excitations: river flow, car, etc. The dynamics of the system is the way the system evolves and the dynamical model is a set of mathematical laws that describe the system up to certain precision. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 2

3 Dynamical system 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 3

4 Different mathematical models difference equations xn+1 = 5xn + 3xn 2 non-stiff ODEs x = sin(t) + x stiff ODEs x = x 2 (1 x) 0 6 t 6 2/, y(0) = DAEs x = xy + 0 = x2 + y Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 4

5 Stiff ODEs: Combustion processes Moler, Matlab exchange 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 5

DAEs: robot arm right@don Riley, Matlab exchange 13.

6 DAEs: robot arm Riley, Matlab exchange 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 6

7 Well posed ODE The process of solving an ODE x = f (t, x) is a well-posed problem, i.e. 1. A solution exists 2. The solution is unique 3. The solution s behavior changes continuously with the initial conditions, only if Lipschitz condition is satsified kf (x) f (y)k 6 Lkx yk and the ODE is dissipative, i.e. that the two solutions x(t) and y(t) of the same ODE with different initial values x0 and y0 are contractive. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 7

8 Stiff ODE Note that the stiff ODE is a well-posed problem only if a one-sided Lipschitz condition is satisfied: hf (x) f (y), x yi 6 `kx yk2 and the ODE is dissipative, i.e. that the two solutions x(t) and y(t) of the same ODE with different initial values x0 and y0 are contractive. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 8

9 DAEs It is well known that one of the key aspects in which a system of DAEs differs from a system of ODEs is that, to get the solutions of DAEs, only continuity of the right-hand side f (t) may not be sufficient and therefore higher derivatives of f (t) may be required. For linear systems that relies on Theorem Let (A, B) be the regular pair of square matrices and let P and Q be nonsingular matrices which transform original problem into the canonical form such that A x = B x + f (t) Furthermore, let ν be the index of the matrix pair and f be at least ν times differentiable. Then, we have the following: the differential equation is solvable, an initial condition is consistent and every equation with consistent initial condition is uniquely solvable. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 9

10 Numerical integration of ODEs After numerical integration of ODE x = f (x, t) by time step size h and slight mathematical transformations to a form xn+1 = M(xn ) one obtains difference equation. Hence, one may study the following 1. is ODE stable? What are its equilibria points? (ODE1) 2. which numerical scheme to use to integrate the system? 3. is the numerical scheme stable wrt changes in initial condititions? (zero stability) 4. what about the choice of the step-size h to keep the numerical scheme stable? (A,AN,B,BN stability) 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 10

11 2) which numerical scheme to use to integrate the system? 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 11

12 Numerical schemes 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 knowledge on xn multistep methods: these methods require knowledge on more than one time point 3 1 xn+1 = xn + h fn fn 1 requires knowledge on xn, xn Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 12

13 One step methods: IRK General implicit Runge Kutta method is presented by Butcher table: c1 a11 a12 c2 a21 a cs as1 as2 b1 b2... a1s... a2s......,... ass... bs or c A bt Here, A may be such that aij = 0, j > i explicit Runge Kutta method (ERK) such that aij = 0, j > i and at least one aii 6= 0 diagonal implicit Runge Kutta method (DIRK) such that aij = 0, j > i and all aii 6= 0 and identical singly diagonal implicit Runge Kutta method (SDIRK) full matrix fully implicit Runge Kutta method (IRK) 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 13

14 Explicit RK first order Taylor expansion explicit Euler xn = xn 1 + dx t h = xn 1 + hf (tn 1, xn 1 ), dt n 1 xa (tn ) = xn + O(h2 ) higher order Taylor methods xn = xn 1 + dx d 2x (tn tn 1 ) + 2 (tn tn 1 ) dt tn 1 dt tn 1 xa (tn ) = xn + O(hp+1 ) xa (tn ) = x(tn 1 ) + h(b1 k1 + b2 k bs ks ) + O(hp+1 ) the local error loc = xa (tn ) x(tn 1 ) h s X bi f (tn 1 + hci, x(tn 1 + hci )) i=1 satisfies consistency condition maxk loc h 1 k 6 Chp 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 14

15 Example of Runge Kutta method of second 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 15

16 Collocation based RKM To numerically evaluate the integral in Z tn x(tn ) = x(tn 1 ) + f (τ, x)dτ, tn 1 we assume P(tn 1 ) = x(tn 1 ), P(tn 1 + h) = x(tn ). and dp (tn 1 +ci h) = f (tn 1 +ci h, P(tn 1 +ci h)) =: ki, i = 1,..., s dt in which P(t) is the polynomial of degree s N whose derivative dp dt coincides with the function f at the distinct collocation points (tn 1 + ci h) for i = 1,..., s and c1,..., cs being distinct real numbers between 0 and Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 16

17 Order conditions These three equalities can be gathered to B(p) : s X bi cim 1 = i=1 C(q) : s X aij cjm 1 = j=1 D(r) : s X 1, m cim, m bi cim 1 aij = bj i=1 k = 1,..., p bt c m 1 = 1 m i = 1,..., s, m = 1,...q (1 cjm ) m, j = 1,..., s, m = 1,..., r with r=q = s and p = s. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 17

18 Rosenbrock Wanner method The ROW method is derived from the autonomous system of ODEs x = f(x). Namely, by transforming the system to x = Jx + f(x) Jx in which J = fx (x0 ), one may distinguish stiff and linear part of ODE Jx from the non-stiff and nonlinear part of ODE f(x) Jx. Integrating the stiff part by DIRK method (characterised by coefficients aij ), and non-stff by explicit Runge Kutta (characterised by coefficients αij ) one obtains the ROW method (I haii J)ki = f(x0 + h i 1 X j=1 αij kj ) + hj i 1 X (aij αij )kj j=1 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 18

Adaptive Runge Kutta methods The accuracy of Runge Kutta methods can be numerically estimated: by halving step size (Richardson s method) by increasing the order (embedded Runge

19 Adaptive Runge Kutta methods The accuracy of Runge Kutta methods can be numerically estimated: by halving step size (Richardson s method) by increasing the order (embedded Runge Kutta methods) These error estimates then further can help us to control the step size adaptive technique. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 19

20 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 k+1 = x(tn k+1 ) implicit MM: interpolate f (rhs/derivative) by using points xn+1 = x(tn+1 ), xn = x(tn ),,..., xn k+1 = x(tn k+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 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 20 j > 0, j 6 k

21 Backward differentiation formulas Similar to Adams-Moulton formulas with the only difference that the solution is being interpolated, i.e. (tn+1, x(tn+1 )),..., (tk p, x(tk 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 In general, all linear multistep methods can be written as k X i=0 ai xn+i = h k X bi fn+i, i=0 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 21 (1)

22 Comparison: order and efficiency explicit methods order p 6 s but very efficient as one does not need to solve any system of equations DIRK order p = 1 (SDIRK) and p = 2 (EDIRK) but one has to solve nonlinear system though simpler than in case of IRK collocation based IRK order is p = 2s for Gauss-Legendre, p = 2s 1 for Radau and p = 2s 2 for Lobatto. One has to solve nonlinear system. Rosenbrock Wanner order is p if s > p 1. One solves linear system. multistep methods order is p = s (AB, BDF), p = s + 1 (AM). Linear and nonlinear equatons possible. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 22

23 Problem by integrating stiff system xa (tn ) = x(tn ) + O(hp ) + Res(λ, h) in non-stiff case h 0, z = λh 0, h 0, z = λh while in stiff Fro Protero-Robinson the error x(tn + h) xa (tn + h) = R(z)(x(tn ) g(tn )) zbt (I za) 1 n 0,h Stability is clearly necessary R(z) < 1 but it is not sufficient to obtain accurate solutions to stiff systems of ordinary differential equations. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 23

24 Problem by integrating stiff system For stiff system we have to be careful about stage errors n quadrature errors 0,h In other words, our accuracy can be reduced by a low stage order as well as by a low quadrature order. This leads to order reduction. Definition An IRK with a non-singular matrix A which satisfies the following condition asj = bj AT e = b is called stiffly accurate. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 24

25 is the numerical scheme stable wrt changes in initial condititions or time step size? 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 25

26 Zero stability of one and multistep methods To check zero stability, one applies the numerical scheme on x = 0 and obtains the difference equation: s X ai xi = 0 i=1 Taking the ansatz xi = cξi one obtains the characteristic root equation ρ(ξ) = k X ai ξi = 0 i=0 A scheme is zero stable if every root ξi of the first characteristic polynomial has magnitude smaller than one, ξi 6 1, and if every root ξi with ξi = 1 is a simple root of ρ. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 26

27 Stability wrt to the system type and h In order to discuss this point, one may distingusih different cases of systems A stability (linear autonomous system) A := {f : f (t, x) = qx, q C 1 } AN stability (linear non-autonomous system) AN := {f : f (t, x) = q(t)x, q(t) C 1, t R+ } B stability (non-linear autonomous system) B := {f : x, y Rn } (f (x) f (y), x y) 6 0, BN stability (non-linear non-autonomous system) BN := {f : (f (t, x) f (t, y), x y) 6 0, x, y Rn, 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 27 t R+ }

28 A-stability A Runge-Kutta method is called A-stable, if the stability function R0 (z) satisfies R0 (z) 6 1, z C := {z C Re z 6 0}. A Runge-Kutta method is called strongly A-stable, if the RK-method is A-stable and the stability function R0 (z) satisfies lim R0 (z) < 1. z A Runge-Kutta method is called L-stable (left-stability, see Ehle (1973), if the RK-method is stongly A-stable and the stability function R0 (z) satisfies lim R0 (z) = 0. z 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 28

29 A(α) stability Where A-stability is impossible or difficult to achieve, a weaker property is acceptable for the solution of many problems. Definition Let α denote an angle satisfying α (0, π) and let S(α) denote the set of points x + iy in the complex plane such that x 6 0 and tan(α) x 6 y 6 tan(α) x. A Runge Kutta method with stability function R(z) is A(α)-stable if R(z) 6 1 for all z 2008, book 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 29

30 AN Stability We consider the stability properties of the IRK method applied to a single homogeneous linear equation in which the complex valued coefficient is not necessarily constant x = λ(t)x and λ(t) : R C is continuous (to ensure that the problem is well posed). Furthermore, we suppose that λ takes values only with nonpositive real part, to ensure that the ODE has a solution with nonincreasing magnitude. Definition A Runge-Kutta method is AN-stable if its the stability function R(Z satisfies R(Z) = 1 + bt Z(I AZ) 1 e) < 1 for all Z = diag(z1, z2,.., zs ) such that zi = zj whenever ci = cj and such that Re(Z) =< 0 for 1, 2,., s. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 30

31 What about nonlinear equations?ity If the right hand side of the ODE x = f (x), x(t0 ) = x0 is satisfying monotonicity condition hf (x) f (y), x yi 6 0 in which h, i is inner product with corresponding norm k k and x and y two solutions corresponding to the exact and perturbed initial values. Then, the previous condition means the following d kx yk2 6 0 dt which shows that kx yk does not increase with time. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 31

32 BN stability Definition A Runge Kutta method applied to the non-autonomous non-linear system dx = f (t, x) dt which verifies dissipative condition hf (t, y) f (t, z), y zi < 0 is called BN-stable if this condition implies kxn+1 yn+1 k 6 kxn yn k for two numerical solutions x and y starting from two different initial conditions. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 32

33 DAEs 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 33

34 Differential algebraic equations The problem described 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. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 34

35 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. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 35

36 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 kx (t) x(t)k Zt 6 C kx (0) x(0)k + sup τ I δ(τ) dτ + t0 ip X 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. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 36

37 Index-1 and Index-2 problems We have focused on two groups of DAEs classified by the value of differentiation index to Index-1 problems y = f (t, y, x), y(t0 ) = y0 0 = g(t, y, x) if gx is invertible. Index-2 problems y = f (t, y, x), y(t0 ) = y0 0 = g(t, y) if gy fx is invertible. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 37

Numerical integration of DAEs Numerical methods for DAE can be classified to direct discretisation index reduction techniques The first one does not require reformulation (differentiation), whereas

38 Numerical integration of DAEs Numerical methods for DAE can be classified to direct discretisation index reduction techniques The first one does not require reformulation (differentiation), whereas the second does. However, the problem is that the direct discretisation can only be used for special classes of systems such as index-1 and index-2 problems. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 38

39 Numerical solution of Index-1 problem To numerically solve the index-1 DAE one may use one of the following techniques: state-space form method x = G(t, y) y = f (t, y, G(t, y)), y(t0 ) = y0 ε-embedding method y = f (t, y, x), y(t0 ) = y0 εx = g(t, y, x) index reduction method 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 39

40 Problems Methods often exhibit order reduction as they do when solving stiff ODEs. A major practical difference between numerically solving ODEs and DAEs is that the DAE system must be initialized with values that are consistent with all the constraints even the hidden ones! It can be shown that the condition number of the Newton iteration matrix is O(h id ). For small h, the Newton iteration might fail. Hence, one would have to use projection or stabilisation algorithms to prevent the dirft offs. Basically obtained ODE of first order can become unstable linearly or even exponentially 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 40

41 Projected Runge Kutta method To prevent the dirft off, one may project the solution back to the manifold defined by the original constraint. Let (xn 1, yn 1 ) be the point consistent to the system. After one step of RK we obtain x n and y n which is then projected back onto the original manifold. One way of doing this is to minimise minky n yn k yn under constraint g(yn ) = Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 41

42 Baumgarte stabilisation Let us observe the constraint g(y) = 0 and differentiate with respect to time g = gy f = 0 To prevent drift off one would like to satisfy the first two equations in time when using numerical procedure. However, this is usually not achieved with classical numerical method. Thus, Baumgarte came to idea to combine them into g + αg = 0 and take it as a new constraint instead of g(y) = Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 42

Hamiltonian systems The Hamiltonian system is defined by the total energy formulated as H = Ek (p) + Ep (q), in which H is the Hamiltonian, and Ek and Ep denote the kinetic and potential energy,

43 Hamiltonian systems The Hamiltonian system is defined by the total energy formulated as H = Ek (p) + Ep (q), in which H is the Hamiltonian, and Ek and Ep denote the kinetic and potential energy, respectively. Here, (p, q) are so-called canonical coordinates of the mechanical system in which p is the generalised momentum and q are generalised coordinates. Given so, one may model the mechanical system in a following form p k = H, qk q k = having that H(p, q) = const =: c H pk t 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 43

44 Symlectic Runge Kutta method The Runge Kutta method ki = f (tn + ci h, yn + h s X aij kj ), i = 1,.., s j=1 yn+1 = yn + h s X bi ki i=1 is symplectic if it conserves quadratic first integrals I(yn+1 ) = I(yn ) whenever I(y) = y T Cy is a first integral of y = f (y). Here, C is the symmetric matrix. One may show that this holds when the coefficients of Runge Kutta method satisfy the relation bi aij + bj aji = bi bj, i, j = 1,..., s Example of these methods are Gauss collocation methods. 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 44

45 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 45

46 Is it the end?! No. This is just beginning. The universe may be timeless, but if you imagine breaking it into pieces, some of the pieces can serve as clocks for the others. Time emerges from timelessness. We perceive time because we are, by our very nature, one of those pieces. Craig Callender Scientific American June 2010 Thank you for following lectures and good luck! 13. Juli 2016 Bojana Rosic Advanced methods for ODEs and DAEs Seite 46

Advanced methods for ODEs and DAEs

Advanced methods for ODEs and DAEs Lecture 8: Dirk and Rosenbrock Wanner methods Bojana Rosic, 14 Juni 2016 General implicit Runge Kutta method Runge Kutta method is presented by Butcher table: c1 a11