Stabilität differential-algebraischer Systeme

Transcription

1 Stabilität differential-algebraischer Systeme Caren Tischendorf, Universität zu Köln Elgersburger Arbeitstagung, Februar 2008 Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

2 Outline 1 Stability Observation for DAEs - Circuit Example 2 What does A-stability mean? 3 Test Equation for DAEs? 4 Numerical Analysis for DAEs Reformulated 5 Concepts of B- and G-stability for ODEs/DAEs 6 Open Questions Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

3 Stability Observation for DAEs - Circuit Example Test Circuit C 1 2 i (j,t) S V G (t) 1 2 G (t) 1 j V C 2 v(e,t) 2 Circuit Equations ė 1 + G 1 (t)e 1 + j V = 0 ė 2 + G 2 (t)e 2 + i s (j V, t) = 0 e 1 = v(e 2, t) G 1(t) = 1 ηt λ, G 2(t) = η 1 ηt λ ηt v(e 2, t) = 1 1 ηt e2 i s(j V, t) = ηtj v e 1, e 2 - node potentials j V - current through the voltage source Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

4 Stability Observation for DAEs - Circuit Example Solution with the implicit Euler method for parameters λ = 5, η = 20 and stepsize h = 0.02 Stable solution! Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

5 Stability Observation for DAEs - Circuit Example Solution with the implicit Euler method for parameters λ = 5, η = 20 and stepsize h = 0.06 Unstable solution! Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

6 Stability Observation for DAEs ODE exact solution is asymptotically stable A-stable method numerical solution is asymptotically stable DAE exact solution is asymptotically stable A-stable method numerical solution is asymptotically stable reported for the first time in U. Ascher, L. R. Petzold. Stability of computational methods for constraint dynamic systems. SIAM J. Sci. Comp. 14, pp , Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

7 A-Stability A numerical method is A-stable. Dahlquists test equation The numerical method provides asymptotically stable solutions for x = λx, Re λ < 0. The numerical method provides asymptotically stable solutions for x = Ax, Re ϱ(a) < 0.. Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

8 A-Stability A numerical method is A-stable. Dahlquists test equation The numerical method provides asymptotically stable solutions for x = λx, Re λ < 0. The numerical method provides asymptotically stable solutions for x = Ax, Re ϱ(a) < 0. Is there a test equation for DAEs? How should it look like?. Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

9 Numerical Analysis for a Second DAE Example DAE (λ 1)x 1 + λtx 2 = 0 (λ 1)x 1 + (λt 1)x 2 = 0 inherent ODE + constraint x 2 = λx 2 x 1 = λt 1 1 λ x 2 Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

10 Numerical Analysis for a Second DAE Example DAE (λ 1)x 1 + λtx 2 = 0 (λ 1)x 1 + (λt 1)x 2 = 0 numerical method implicit Euler for DAE (λ 1) x 1,n x 1,n 1 h + λt n x 2,n x 2,n 1 h = 0 (λ 1)x 1,n + (λt n 1)x 2,n = 0 inherent ODE + constraint x 2 = λx 2 x 1 = λt 1 1 λ x 2 Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

11 Numerical Analysis for a Second DAE Example DAE (λ 1)x 1 + λtx 2 = 0 (λ 1)x 1 + (λt 1)x 2 = 0 numerical method implicit Euler for DAE (λ 1) x 1,n x 1,n 1 h + λt n x 2,n x 2,n 1 h = 0 (λ 1)x 1,n + (λt n 1)x 2,n = 0 inherent ODE + constraint x 2 = λx 2 x 1 = λt 1 1 λ x 2 explicit Euler for inherent ODE + constraint x 2,n = (1 + λh)x 2,n 1 x 1,n = λt n 1 1 λ x 2,n Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

12 Analysis of the Numerical Behaviour for Index-1 DAEs Assumption: P is a smooth projector that spans ker E DAE Ex = g(x, t) inherent ODE + constraint (Px) = P (Px) + Pf (Px, t) Qx = Qf (Px, t) Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

13 Analysis of the Numerical Behaviour for Index-1 DAEs Assumption: P is a smooth projector that spans ker E DAE Ex = g(x, t) numerical method inherent ODE + constraint (Px) = P (Px) + Pf (Px, t) Qx = Qf (Px, t) BDF method for DAE E n ρx n = g(x n, t n ) Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

14 Analysis of the Numerical Behaviour for Index-1 DAEs Assumption: P is a smooth projector that spans ker E DAE Ex = g(x, t) numerical method inherent ODE + constraint (Px) = P (Px) + Pf (Px, t) Qx = Qf (Px, t) BDF method for DAE E n ρx n = g(x n, t n ) some method for inherent ODE + constraint ρ(px) n = ρ(px) n P n ρx n + P n f ((Px) n, t n ) Q n x n = Q n f ((Px) n, t n ) Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

15 Analysis of the Numerical Behaviour for Index-1 DAEs Assumption: P is a smooth projector that spans ker E DAE Ex = g(x, t) inherent ODE + constraint (Px) = P (Px) + Pf (Px, t) Qx = Qf (Px, t) numerical method if im P = ker E is constant BDF method for DAE E n ρx n = g(x n, t n ) BDF method for inherent ODE + constraint (Px) n = ρ(px) n P n ρx n + P n f ((Px) n, t n ) Q n x n = Q n f ((Px) n, t n ) Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

16 Test Equation for DAEs? P. Kunkel, V. Mehrmann. Stability properties of differential-algebraic equations and spin-stabilized discretisations. Electronic Transactions on Numerical Analysis (ETNA) 26, pp , Proposal for a test equation for DAEs ( ) ( ) 1 ωt x λ ω(1 λt) = x ωt Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

17 Test Equation for DAEs? P. Kunkel, V. Mehrmann. Stability properties of differential-algebraic equations and spin-stabilized discretisations. Electronic Transactions on Numerical Analysis (ETNA) 26, pp , Proposal for a test equation for DAEs ( ) ( ) 1 ωt x λ ω(1 λt) = x ωt Time dependent transformation y = Rx { Ey = Ay y 1 = λy 1 y 1 = y 2 with ( ) ( ) ( ) 1 0 λ 0 1 ωt E =, A =, R = Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

18 Test Equation for DAEs? P. Kunkel, V. Mehrmann. Stability properties of differential-algebraic equations and spin-stabilized discretisations. Electronic Transactions on Numerical Analysis (ETNA) 26, pp , Proposal for a test equation for DAEs ( ) ( ) 1 ωt x λ ω(1 λt) = x ωt The stability domain depends on λ (asymptotical behaviour of the inherent ODE) and on ω (measuring the time dependent movement of the derivative term of the DAE). Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

19 Test Equation for DAEs? P. Kunkel, V. Mehrmann. Stability properties of differential-algebraic equations and spin-stabilized discretisations. Electronic Transactions on Numerical Analysis (ETNA) 26, pp , Proposal for a test equation for DAEs ( ) ( ) 1 ωt x λ ω(1 λt) = x ωt The stability domain depends on λ (asymptotical behaviour of the inherent ODE) and on ω (measuring the time dependent movement of the derivative term of the DAE). Open question: if a (spin stabilized) method is stable for the test equation, is it also stable for all linear (or at least a certain class of linear) DAEs? Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

20 Test Equation for DAEs? J. Wensch, R. Weiner, K. Strehmel. Stability investigations for index-2 systems. Technical Report, Univ. Halle, Hessenberg Index 2 DAE with a... y = By + Cz + q 0 = Dy + r... Scalar Essential Underlying ODE v = (RBS + R S)v + Rq (R + RB)Fr with RS = 0, RF = 0, DS = 0, DF = I, SR + FD = I, F = C(DC) 1 RBS = const, R S = const, R = const Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

21 Test Equation for DAEs? J. Wensch, R. Weiner, K. Strehmel. Stability investigations for index-2 systems. Technical Report, Univ. Halle, Hessenberg Index 2 DAE with a... y = By + Cz + q 0 = Dy + r A-stability for certain projected Runge-Kutta methods... Scalar Essential Underlying ODE v = (RBS + R S)v + Rq (R + RB)Fr with RS = 0, RF = 0, DS = 0, DF = I, SR + FD = I, F = C(DC) 1 RBS = const, R S = const, R = const Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

22 Stability Observation for DAEs - Third Example DAE of Index 2 [Gear, Petzold 86] x 1 + ηtx 2 + (1 + η)x 2 = q 1 x 1 + ηtx 2 = q 2 Unique solution x 2 = q 1 q 2 x 1 = q 2 ηtx 2 Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

23 Stability Observation for DAEs Numerical solution for different methods and parameter values All tested methods get into trouble for certain DAEs and stepsizes. Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

24 Stability Observation for DAEs - Third Example DAE of Index 2 DAE of Index 2 Reformulated x 1 + ηtx 2 + (1 + η)x 2 = q 1 x 1 + ηtx 2 = q 2 x 1 + (ηtx) 2 + x 2 = q 1 x 1 + ηtx 2 = q 2 Unique solution x 2 = q 1 q 2 x 1 = q 2 ηtx 2 Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

25 Stability Observation for DAEs Reformulated Numerical solution for different methods and parameter values All methods work without any trouble. Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

26 Numerical Analysis for the Second DAE Example Reformulated original DAE (λ 1)x 1 + λtx 2 = 0 (λ 1)x 1 + (λt 1)x 2 = 0 Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

27 Numerical Analysis for the Second DAE Example Reformulated original DAE (λ 1)x 1 + λtx 2 = 0 (λ 1)x 1 + (λt 1)x 2 = 0 DAE reformulated (λ 1)x 1 + (λtx 2 ) λx 2 = 0 (λ 1)x 1 + (λt 1)x 2 = 0 Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

28 Numerical Analysis for the Second DAE Example Reformulated DAE (λ 1)x 1 + (λtx 2 ) λx 2 = 0 (λ 1)x 1 + (λt 1)x 2 = 0 Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

29 Numerical Analysis for the Second DAE Example Reformulated DAE (λ 1)x 1 + (λtx 2 ) λx 2 = 0 (λ 1)x 1 + (λt 1)x 2 = 0 inherent ODE + constraint x 2 = λx 2 x 1 = λt 1 1 λ x 2 Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

30 Numerical Analysis for the Second DAE Example Reformulated DAE (λ 1)x 1 + (λtx 2 ) λx 2 = 0 (λ 1)x 1 + (λt 1)x 2 = 0 numerical method implicit Euler for DAE (λ 1) x 1,n x 1,n 1 λx 2,n h + λt nx 2,n λt n 1 x 2,n 1 h = 0 (λ 1)x 1,n + (λt n 1)x 2,n = 0 inherent ODE + constraint x 2 = λx 2 x 1 = λt 1 1 λ x 2 Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

31 Numerical Analysis for the Second DAE Example Reformulated DAE (λ 1)x 1 + (λtx 2 ) λx 2 = 0 (λ 1)x 1 + (λt 1)x 2 = 0 numerical method implicit Euler for DAE (λ 1) x 1,n x 1,n 1 λx 2,n h + λt nx 2,n λt n 1 x 2,n 1 h = 0 (λ 1)x 1,n + (λt n 1)x 2,n = 0 inherent ODE + constraint x 2 = λx 2 x 1 = λt 1 1 λ x 2 implicit Euler for inherent ODE + constraint x 2,n = 1 1 λh x 2,n 1 x 1,n = λt n 1 1 λ x 2,n Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

32 Analysis of the Numerical Behaviour for Index-1 DAEs Assumption: R is a smooth projector that spans ker A im D DAE A(Dx) + g(x, t) = 0 inherent ODE + constraint (Dx) = R (Dx) + Rf (Dx, t) Qx = Qf (Dx, t) Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

33 Analysis of the Numerical Behaviour for Index-1 DAEs Assumption: R is a smooth projector that spans ker A im D DAE A(Dx) + g(x, t) = 0 inherent ODE + constraint (Dx) = R (Dx) + Rf (Dx, t) Qx = Qf (Dx, t) numerical method BDF method for DAE A n ρ(dx) n + g(x n, t n ) = 0 Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

34 Analysis of the Numerical Behaviour for Index-1 DAEs Assumption: R is a smooth projector that spans ker A im D DAE A(Dx) + g(x, t) = 0 inherent ODE + constraint (Dx) = R (Dx) + Rf (Dx, t) Qx = Qf (Dx, t) numerical method BDF method for DAE A n ρ(dx) n + g(x n, t n ) = 0 some method for inherent ODE + constraint R n ρ(dx) n = R n f ((Dx) n, t n ) Q n x n = Q n f ((Dx) n, t n ) Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

35 Analysis of the Numerical Behaviour for Index-1 DAEs Assumption: R is a smooth projector that spans ker A im D DAE A(Dx) + g(x, t) = 0 inherent ODE + constraint (Dx) = Rf (Dx, t) Qx = Qf (Dx, t) numerical method if im D or ker A is constant BDF method for DAE A n ρ(dx) n + g(x n, t n ) = 0 BDF method for inherent ODE + constraint ρ(dx) n = R n f ((Dx) n, t n ) Q n x n = Q n f ((Dx) n, t n ) Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

36 Analysis of the Numerical Behaviour for Index-1 DAEs Assumption: R is a smooth projector that spans ker A im D DAE A(Dx) + g(x, t) = 0 inherent ODE + constraint (Dx) = Rf (Dx, t) Qx = Qf (Dx, t) numerical method if im D or ker A is constant BDF method for DAE A n ρ(dx) n + g(x n, t n ) = 0 BDF method for inherent ODE + constraint ρ(dx) n = R n f ((Dx) n, t n ) Q n x n = Q n f ((Dx) n, t n ) Such a reformulation is always possible if E is continuous with constant rank for Ex = h(x, t). Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

37 Analysis of the Numerical Behaviour for DAEs The diagram DAE numerical method for DAE inherent ODE + constraint (same) numerical method for inherent ODE + constraint commutes for BDF and stiffly accurate Runge-Kutta methods for index-1 DAEs if the DAE is formulated as A(Dx) + g(x, t) = 0, ker A im D, ker A or im D constant for higher index DAEs more restrictions are needed [Higueras, März, T. 03], [Lamour, März, T. in prep.] Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

38 Concepts of B- and G-stability for ODEs Wish: If x(t) y(t) x(t 0 ) y(t 0 ), t > t 0, for any two solutions of x (t) = f (t, x(t)) then X n Y n X n 1 Y n 1 for numerical solutions X n = (x n,..., x n k+1 ) and Y n = (y n,..., y n k+1 ). class of ODEs x = λx, Reλ < 0 x = f (t, x), f (t, x) f (t, y), x y β x y 2 x = f (t, x), f (t, x) f (t, y), x y β x y 2 numerical method A-stable method B-stable Runge-Kutta G-stable linear multist. Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

39 Criteria for A-, B-, G-stability for ODEs G-stability G-stability = A-stability [Dahlquist 78] B-stability Algebraic stability, i.e. B-stability A-stability b i 0, M = (b i a ij + b j a ji b i b j ) ij pos. semi-definite implies B-stability. [Burrage & Butcher 79, Crouzeix 79] Are the criteria for the ODE case also true in the DAE case? Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

40 Reflection of the Asymptotic Behavior for DAEs Wish (?): If x(t) y(t) x(t 0 ) y(t 0 ) for any two solutions of A(Dx) + g(x, t) = 0 then X n Y n X n 1 Y n 1 for numerical solutions X n = (x n,..., x n k+1 ) and Y n = (y n,..., y n k+1 ). Should we require this behaviour for the whole solution component? Is it appropriate to require such a solution behavior for the dynamic components only? Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23

41 Conclusions and Open Questions A-stable methods may lead to stepsize restrictions (for stability reasons) in case of DAEs applicable range of test equations for DAEs not clear proper reformulations (cheap) of the DAE can avoid such stepsize restrictions concepts as B- and G-stability need criteria for contractivity of DAEs contractivity conditions for the dynamic components are developped for index-1 DAEs contractivity in the higher index DAEs? B- and G-stability for DAEs? contractivity for all components? contractivity for dynamic components in a certain subspace? contractivity/boundedness of the constraints? Tischendorf (Univ. zu Köln) Stabilität von DAEs Elgersburg, / 23