[4-7] Rosenbrock ( ) Runge-Kutta(RK)

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Ἀλέξανδρος Αλεβιζόπουλος
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1 CHINESE JOURNAL OF ENGINEERING MATHEMATICS Vol. 28 No. 3 June 2011 : ) BDF, ) : 1- BDF) 2- : 2- : AMS2000) 90C29 : O221.4 : A 1 ) [1,2] Petzold [3] ) ) [4-7] Rosenbrock ) Runge-KuttaRK) θ- BDF ) ) ) [4,5,8,9] θ- ) 1- [10,11] [12-14] RK 1-2- [15] 1-2- [16,17] RK 1- BDF) 2- : : ).. : ) ) 09JJ3002) 11QDZ01).

2 DDAEs) y x) = f yx), yx τx)), zx) ), x [0, T ], 0 = g yx) ), x [0, T ], z0) = z 0, yx) = ϕx), x [ τ, 0], 1) τx) 0 < τx) τ, 0 < τ x) < 1 f : R n1 R n1 R n2 R n1, g : R n1 R n2 ϕx) g y y)f z y, yx τx)), z) 1) yx), zx) p BDF 1) k α i y n+i = hf y n+k, yn, h ) z n+k, 0 = gyn+k ), 2) x n+k = x n + kh, n 0 ϕ x n+k τx n+k ) ), x n+k τx n+k ) 0, yn h = q Q j δ n )y n+k mn+j, x n+k τx n+k ) > 0, j= u τx n+k ) = m n δ n )h, u, q, m n Z +, δ n [0, 1), q + u = p, q + 1 m n, Q j δ n ) 2) 1 k α i ŷ n+i = hf ŷ n+k, ŷn, h ) ẑ n+k + hδ, 0 = gŷn+k ) + θ. 4) y n+k, z n+k 2) ŷ n+k, ẑ n+k 4) 3) y n+j yx n+j ) = Oh), z n+j zx n+j ) = Oh), gy n+j ) = Oh 2 ), j = 0, 1,, k 1, x n+j = x n + jh, 5) ŷ n+j y n+j = Oh 2 ), ẑ n+j z n+j = Oh), δ = Oh), θ = Oh 2 ), 6) h < h 0 ŷ n+k y n+k C Ŷn Y n + h δ + θ ), 7) ẑ n+k z n+k C h k 1 gy ŷ n+k )ŷ n+j y n+j ) ) + h Ŷ n Y n + h δ + θ, 8) Ŷ n Y n = ŷn+k 1 T yn+k 1, T, ŷn T yn T, ŷn) h T yn) h T ) T, 9) Ŷn Y n = max max ŷt n+j yn+j T, ŷ h n ) T yn) h T ). 10) 0 j k 1

3 3 2- BDF 337 k 1 α i η = y n+i, α k h δ 2) 4) k 1 α i ˆη = ŷ n+i. 11) α k 12) y n+k = η + hfy n+k, y h n, z n+k ), 0 = gy n+k ), 12) ŷ n+k = ˆη + hfŷ n+k, ŷ h n, ẑ n+k ) + hδ, 0 = gŷ n+k ) + θ. 13) 0 = gy n+k ) gη) + gη) = 12) 14) 1 13) g y η + τyn+k η) ) dτ[y n+k η] + gη), 14) g y η + τyn+k η) ) dτfy n+k, yn, h z n+k ) + 1 gη) = 0, 15) h g y ˆη + τŷn+k ˆη) ) dτ [ fŷ n+k, ŷn, h ẑ n+k ) + δ ] + 1 h gˆη) + 1 θ = 0. 16) h f, g g y f z 15)-16) ẑ n+k z n+k C 1 ŷ n+k y n+k + ˆη η + ŷ h n y h n + δ + 1 h θ + 1 ), h gˆη) gη) 17) 12) 13) ŷ n+k y n+k ˆη η + h fŷn+k, ŷn, h ẑ n+k ) fy n+k, yn, h z n+k ) ˆη η + hl ŷ n+k y n+k + ŷn h yn h + ẑ n+k z n+k ), 18) L f Lipschitz 17) 18) ŷ n+k y n+k C 2 ˆη η + ŷ h n y h n + h δ + θ ), 19) 17) h 1 L + LC 1. 19) ẑ n+k z n+k C 3 ˆη η + ŷ h n y h n + δ + 1 h θ + 1 ) h g 1 yˆη)ˆη η), h, 20) L + LC 1 k 1 α j ˆη η = ŷ n+j y n+j ) C 4 Ŷn Y n, 21) α k gy ˆη)ˆη η) gy ŷ k )ˆη η) + ˆη η Oh) k 1 C 5 gy ŷ k )ŷ n+j y n+j ) ) + Ŷ n Y n Oh). 22)

4 ), 22) 19), 20) 7), 8) 1 2-DDAEs 1) g y f z p 3) BDF 2) p y k yx k ) = Oh p+1 ), z k zx k ) = Oh p ). 23) 1 n = 0, ŷ j = yx j ), ẑ j = zx j ), j = 0, 1,, k 1 δ = Oh p ), θ = 0 3) ŷ0 h y0 h = Oh p+1 ) DDAEs 1) g y f z k 6 2) y j yx j ) = Oh p+1 ), j = 0, 1,, k 1, 24) p 3) k BDF 2) p = k y n yx n ) = Oh p ), z n zx n ) = Oh p ), x n = nh, n k. 25) y {yn, 0 zn} 0 {yn, l zn}, l l = 1, 2, yj l = yx j), zj l = zx j), j = l 1,, l + k 2 yn l yn l+1 {ŷ n, ẑ n } {ỹ n, z n } 1 3 Ĉ0, Ĉ 1, Ĉ 2 3 ) ŷ n+j yx n+j ) Ĉ0h, ẑ n+j zx n+j ) Ĉ1h, ŷ n+j ỹ n+j Ĉ2h 2, j = 0, 1,, k 1. 26) z n+k = z n+k ẑ n+k, y n+j = ỹ n+j ŷ n+j, j = 0, 1,, k, y h n = ỹ h n ŷ h n, Y n = y T n+k 1,, yt n, y h n) T ) T 1 δ = 0, θ = 0 1 y n+k C Y n ), z n+k C h ) gy ŷ n+k ) y n+j + h Yn. 27) k 1 h C Ĉ0, Ĉ 1, Ĉ 2 3) 26) y n+k = Oh 2 ), z n+k = Oh). 28) k α i y n+i = h [ fỹ n+k, ỹn, h z n+k ) fŷ n+k, ŷn, h ẑ n+k ) ] = hf z ŷ n+j, ŷn, h ẑ n+j ) z n+j + O h Y n ), 29) 0 = g y ŷ n+k ) y n+k + O h Y n ). 30) Q n+j = f z g y f z ) 1 g y )ŷ n+j, ŷn, h ẑ n+j ), P n+j = I Q n+j, j = 0, 1,, k Q 2 n+j = Q n+j, Pn+j 2 = P n+j, Q n+j P n+j = P n+j Q n+j = 0, Q n+j+1 = Q n+j +Oh), 31)

5 3 2- BDF 339 P n+k 29) f z g y f z ) 1 30) k α i P n+i y n+i = O h Y n ), Q n+k y n+k = O h Y n ). 32) 3) 0 < τ x) < 1 26) 27) Y n = U n + V n 32) 1 2 yh n+1 = 1 2 yh n + O h Y n ), 33) Pn+k 1 ) T ) T 1 ) ) U n = y n+k 1,, Pn y n, y h T T 2 n, Qn+k 1 ) T ) T 1 ) ) V n = y n+k 1,, Qn y n, y h T T 2 n, U n+1 = A I)U n + O h U n + h V n ), 34) V n+1 = N I)V n + O h U n + h V n ), 35) A = α k 1 α 1 α , N = , α j = α j/α k ρξ) = k α k ξ k [18] A I 1 N I 1 U n Oh) Oh) U n, 36) V n+1 Oh) 1 + Oh) V n 36) U n T 1 λn 1 0 V n 0 λ n 2 T U 0 V 0, 37) λ 1 = 1 + Oh), λ 2 = 1 + Oh) T T = 1 Oh), Oh) 1

6 U n C U 0 + h V 0 ), 38) V n C h U 0 + V 0 ), 39) Y n C 6 Un + V n ) C 7 U0 + V 0 ). 40) U 0, V 0 y U 0 H 0 h p+1, V 0 H 1 h p+1, y 0 H 2 h p+1, 41) 23), 24), 38)-40) 41) y n C 8 h p+1, g y ŷ n+k ) y n+j C 9 h p+1, 42) y n yx n ) n k+1 l=0 y l n y l+1 n C 10 h p. 43) z n y h n k, y n k,, y n 1 1 ŷ i = yx i ), ẑ i = zx i ), δ = Oh p ), θ = 0 1 z n zx n ) C h k gy yx n ))y n j yx n j )) + Oh p ). 44) 42) 43) g y yx n ))y n j yx n j )) = n k+1 l=0 n k+1 l=0 g y yn l + Oh p ))yn j l y l+1 n j ) ) g y yn)y l n j l y l+1 n j ) + Oh 2p+1 ) = Oh p+1 ), zn zx n ) C11 h p. 45) 1 C 10, C 11 26) Ĉ0, Ĉ 1 h C 10 h p 1 Ĉ0, C 11 h p 1 Ĉ1 26) 3 2- y 1x) = 2 y y 1 z + y 1 x 2 ) 2 y 2 x 2 ), 0 x 2, y 2x) = z2 y 2 y 1 y 2 x 2 ) z 3, 0 x 2, 0 = y 1 y 2 + y 2 2, 0 x 2, y 1 0) = 1, y 2 0) = 1, z0) = 1, 46)

7 3 2- BDF ) y 1 x) = e x, y 2 x) = e x, zx) = e x h, y 1, y 2, z x = 2 yerr1h), yerr2h), zerrh) py1h), py2h), pzh) py1h) = ln yerr1h) / ln 2, py2h) = ln yerr2h) / ln 2, pzh) = ln zerrh) / ln 2. yerr10.5h) yerr10.5h) zerr0.5h) BDF BDF2) BDF BDF3) 46) 1 2 1: BDF2 h yerr1 yerr2 zerr py1 py2 pz CPU E E E E E E E E E E E E : BDF3 h yerr1 yerr2 zerr py1 py2 pz CPU E E E E E E E E E E E E [1]. [J]. 2008, 253): Yu Y X, Wen L P, Li S F. Stability analysis of one-leg methods of delayed integro-differential equations[j]. Chinese Journal of Engineering Mathematics, 2008, 253): [2]. [J]. 2005, 221): Ni T Y, Li P X, Li C X. The stablity of neutral linear differential equation[j]. Chinese Journal of Engineering Mathematics, 2005, 221): [3] Petzold L R. Differential-algebraic equations are not ODEs[J]. SIAM Journal on Scientific and Statistical Computing, 1982, 33): [4] Zhu W L, Petzold L R. Asymptotic stability of linear differential-algebraic equations and numerical methods[j]. Applied Numerical Mathematics, 1997, 242): [5] Campbell S L, Linh V H. Stability criteria for differential-algebraic equations with multiple delays and their numerical solutions[j]. Applied Mathematics and Computation, 2009, 2082): [6] Shampine L F, Gahinet P. Delay differential-algebraic equations in control theory[j]. Applied Numerical Mathematics, 2006, 563): [7] Baker C T H, Paul C A H, Tian H. Differential algebraic equations with after-effect[j]. Journal of Computational and Applied Mathematics, 2002, 1401): [8] Xu Y, Zhao J J, Dong S Y, et al. Stability of the Rosenbrock methods for the neutral delay differentialalgebraic equations[j]. Applied Mathematics and Computation, 2005, 1682):

8 [9] Xu Y, Zhao J J, Sui Z N. Stability analysis of θ-methods for neutral multi-delay-integro-differential system[j]. Discrete Dynamics in Nature and Society, 2008, 791): [10]. [J]. 2000, 282): Zhang C J, Cheng W. The stability criteria of implicit midpoint rule for delay differential-algebraic systems[j]. Journal of Huazhong University of Science and Technology, 2000, 282): [11]. ρ, σ)- [J]. 2001, 183): Zhang C J, Liao X Q. Nonlinear stability of ρ, σ)-methods for stiff delay-differential-algebraic systems[j]. Control Theory & Applications, 2001, 183): [12]. [D] Li H Z. The numerical stability and block methods for delay-differential-algebraic systems[d]. Wuhan: Huazhong University of Science and Technology, 2004 [13]. [D] Gan S Q. The numerical analysis for singular perturbation and delay singular perturbation problems[d]. Beijing Institute of Mathematics and System Science, Chinese Academy of Sciences, 2001 [14] Ascher U M, Petzold L R. The numerical solution of delay-differential-algebraic equations of retarded and neutral type[j]. SIAM Journal on Numerical Analysis, 1995, 325): [15] Hauber R. Numerical treatment of retarded differential-algebraic equations by collocation methods[j]. Advances in Computational Mathematics, 1997, 74): [16]. [J]. 2008, 293): Xiao F Y, Zhang C J. Convergence of one-leg methods for a class of variable retarded differential algebraic equations[j]. Journal on Numerical Methods and Computer Applications, 2008, 293): [17] Xiao F Y, Zhang C J. Convergence analysis of Runge-Kutta methods for a class of retarded differential algebraic systems[j]. Acta Mathematica Scientia, 2010, 301): [18] Hairer E, Wanner G. Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems[M]. Berlin: Springer-Verlag, 1996 Convergence of Backward Differentiation Formulas for Index-2 Differential-algebraic Equations with Variable Delay LIU Hong-liang, XIAO Ai-guo School of Mathematics and Computational Science, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan ) Abstract: Delay differential-algebraic equations often arise in automatic control, power and circuit analysis, multi-body dynamics, etc. The current researches on numerical analysis for delay-differentialalgebraic equations are mainly focused on linear problems and 1-index problems. It is difficult to do numerical analysis for high-index nonlinear delay-differential-algebraic equations, and there are only a few results for this kind of problems, furthermore, most of them are about constant-delay problems. The backward differentiation formulas are applied in this paper to index-2 nonlinear differential-algebraic equations with variable delay. The corresponding convergence results are obtained and confirmed by some numerical examples. Keywords: index-2 differential-algebraic equations; backward differentiation formulas; convergence; variable delay Received: 05 Aug Accepted: 14 Dec Foundation item: The National Natural Science Foundation of China ); the Specialized Research Fund for the Doctoral Program of Higher Education of China ); the Natural Science Foundation of Hunan Province 09JJ3002); the Doctoral Research Fund of Xiangtan University 11QDZ01).

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