DSpace at VNU: On BDF-Based Multistep Schemes for Some Classes of Linear Differential-Algebraic Equations of Index at Most 2

DSpace at VNU: On BDF-Based Multistep Schemes for Some Classes of Linear Differential-Algebraic Equations of Index at Mo...

Acta Math Vietnam DOI 10.1007/s40306-016-0171-2 On BDF-Based Multistep Schemes for Some Classes of Linear Differential-Algebraic Equations of Index at Most Mikhail Valeryanovich Bulatov1,2 · Vu Hoang Linh3 · Liubov Stepanovna Solovarova1 Received: May 2015 / Accepted: 29 November 2015 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016 Abstract A family of efficient multistep difference schemes for solving some classes of linear non-autonomous differential-algebraic equations of index at most is proposed It is shown that if the popular backward differentiation formulas (BDFs) are applied to a reformulated form of the original problem, then the methods preserve the stability property and the convergence order that the corresponding BDF methods possess in the ODE case Further issues such as numerical differentiation that may be involved in the implementation and computational errors are also discussed Finally, several numerical experiments are given which confirm the theoretical results Keywords Linear differential-algebraic equations · Index · Strangeness-free form · Multistep difference schemes · BDF methods · Convergence · Stability function Mathematics Subject Classification (2010) 65L07 · 65L80 Mikhail Valeryanovich Bulatov mvbul@icc.ru Vu Hoang Linh linhvh@vnu.edu.vn Liubov Stepanovna Solovarova soleilu@mail.ru Matrosov Institute of System Dynamics and Control Theory, Siberian Branch, Russian Academy of Sciences, Lermontov St., 134, Irkutsk, Russia Irkutsk National Research Technical University, Lermontov St., 83, Irkutsk, Russia Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam M V Bulatov et al Introduction Coupled systems of differential and algebraic equations often occur as mathematical models in various areas of science and engineering, for instance, in multibody mechanics, electrical networks, chemical processes, and optimal control Real-life examples of such problems can be found in [1, 3, 4, 6, 14, 17] If the equations are linear, setting them in one system, we obtain a system of the form A(t)x (t) + B(t)x(t) = f (t), t ∈ [t0 , tf ], (1) where A and B are n by n matrix functions, f is a n−dimensional vector function, and x is the unknown vector function We assume that the data functions A, B, and f are sufficiently smooth and det A(t) ≡ Such systems are called linear differential-algebraic equations (DAEs) Without loss of generality, we assume in addition that [t0 , tf ] = [0, 1] For DAE (1), we assign initial condition x(0) = x0 , (2) that is supposed to be consistent with the right hand side of (1) A continuously differentiable vector-function that satisfies (1) pointwise for t ∈ [0, 1] and also the initial condition (2) is called a solution of the initial value problem (1)–(2) It is known that the qualitative theory and the numerical analysis of DAEs are more difficult than ODEs, see, e.g., [6, 12, 14, 17, 19] Difficulties that arise with DAEs are characterized by the notion of index This notion can be introduced in several ways such as differentiation index [1, 6], tractability index [11, 19], perturbation index [12, 14], strangeness index [17], and regularization index [9] For example, a DAE is said to have differentiation index r if r is the minimal number of necessary differentiations applied to the DAE in order to get an ODE system having the same solution If the entries of matrices A and B are real analytical functions, and the DAE (1) has differentiation index r, then it is known, see [6, 9], that there exist matrix functions P and Q nonsingular for every t ∈ [0, 1] such that P (t)A(t)Q(t) = Id 0 N (t) P (t)A(t)Q (t) + P (t)B(t)Q(t) = (3) , J (t) 0 In−d (4) , where Id denotes the identity matrix of dimension d, N is an upper triangular matrix with zeros on the diagonal and nilpotent of index r ≥ 1, i.e., N r (t) ≡ 0, N r−1 (t) ≡ for t ∈ [0, 1] The form (3)–(4) is known as the standard canonical form for linear time-varying DAEs, see [2, 6] We note that here we not make a constant rank restriction on A(t) for t ∈ [0, 1], i.e., the rank of A(t) may be varying Furthermore, the index is a global notion Thus, restricting on some sub-interval, the index may be decreased, i.e., the index may vary, as well Multiplying (1) by matrix P (t) and substituting Q(t)y(t) for x(t), we obtain P (t)A(t)Q(t)y + (P (t)A(t)Q (t) + P (t)B(t)Q(t))y(t) = Id 0 N (t) y1 (t) y2 (t) + J 0 In−d y1 (t) y2 (t) = ϕ(t) ψ(t) , (5) On BDF-Based Multistep Schemes for DAEs ϕ(t) = P (t)f (t) Thus, y1 is a solution of the so-called ψ(t) underlying ODE y1 + J (t)y1 = ϕ(t), and y2 is calculated using the following rule, see [4], where (y1T (t), y2T (t))T = y(t), r−1 y2 (t) = (−1)i T i ψ(t), i=0 where the effect of the operator T on a vector-function ψ is defined by T ψ(t) = N (t) d ψ(t), T ψ(t) = ψ(t) dt In literature, there are various approaches to analytical and numerical investigations of DAEs, see, e.g., [4–6, 8, 10–12, 14, 17, 19, 20] The motivation of this work arises from the fact that a number of implicit difference schemes developed for stiff ODEs may be unstable (in the sense of zero-stability or absolute stability, see [1, 13, 14]) for DAEs or singular systems of linear algebraic equations (SLAEs) may arise as the result of discretizations, see, e.g., [6, 9, 14, 18, 21] In addition, an ODE method usually suffers order reduction when it is applied directly to higher-index DAEs For illustration, we show the ineffectiveness of some well-known difference schemes when they are applied to DAEs Example ([9]) The DAE t 0 u v + α t u v = ϕ(t) ψ(t) (6) , where α is a parameter, α = 1, has index The DAE has the unique solution u(t) = ψ(t) − tv(t), v(t) = (ϕ(t) − ψ (t))/(α − 1), which does not depend on the initial data t Here, matrix P (t) = I , matrix Q(t) = −1 If for solving the index-2 DAE (6) with initial condition u(0) = ψ(0), v(0) = (ϕ(0) − ψ (0))/(α − 1), we apply the implicit Euler method, then we obtain ui+1 = ψi+1 − ti+1 vi+1 , vi+1 = α ϕi+1 + vi − ψi+1 − ψi h , where ti = ih, i = 0, 1, , M, h = 1/M, ψi = ψ(ti ), ϕi = ϕ(ti ), ui ≈ u(ti ), vi ≈ v(ti ) Thus, when |α| < 1, we obtain an unstable numerical solution (the error grows at velocity of geometric progression with rate 1/ |α|), and when α = 0, we have a singular SLAE on every step of integration If for this equation we apply a number of other implicit RungeKutta methods, then we can find values of parameter α such that the resulted scheme is unstable or we have a singular SLAE For example, applying the Lobatto III-C two-stage method to (6), if α = −1, then we obtain a singular SLAE Usually difference schemes that are appropriate for stiff ODEs can be applied for numerical solution of DAEs of index However, recently, examples have been constructed [18, 21] that show that these methods are not always stable Namely, considerable restriction of stepsize of integration is required M V Bulatov et al Example ([18]) The IVP −αt 0 u v = λ α(1 − λt) −1 + αt u v , u(0) = v(0) = 1, (7) where λ and α are scalar parameters, has the unique solution u(t) = (1 + αt) exp λt, v(t) = exp λt The DAE has index and it is called the test equation for DAEs which generalizes the classical test equation for ODEs y = λy When λ 1, i.e., the method is unstable in the sense of absolute stability Thus, the implicit Euler method is not unconditionally stable as in the ODE case Though nowadays the numerical analysis of DAEs has become a mature topic [6, 14, 17, 19], these examples show the necessity of developing stability- and order-preserving difference schemes for numerical solution of DAEs In this paper, we will construct a family of stable difference schemes based on the classical BDF methods for a class of linear DAEs of index at most When discretizing, we take into account the structure of the problem The idea is that, instead of discretizing directly the original problem, we will that for a reformulated DAE which is easily obtained from the original DAE The obtained convergence result extends and complements some recent convergence results for low-order methods in [7] Not only the order, but also the stability of the BDF methods are preserved The fact that discretizing appropriately formulated DAEs leads to stability-preserving methods has already been shown by using the projector-based approach for regular index-2 DAEs in [15, 16] and quasi-regular DAEs with harmless critical points in [10] The approach used in this paper sounds similar to that in [10, 15, 16] since discretizations are applied to a reformulated form of the original DAE However, we emphasize the key difference in our study that, for DAEs with index at most 2, the rank of A may be varying and the index may be switching between and 2, but we not consider DAEs in properly stated form In our approach, DAEs that are discretized not necessarily satisfy the assumptions for the numerically qualified form defined in those references Consequently, the convergence results for index2 DAEs obtained in this paper are different from those in [10, 15, 16] For the analysis, we also avoid the use of projectors Another approach for improving the stability is suggested in [18] The advantages of the methods proposed in this paper are that they are easily implemented and efficiently solve stiff linear DAEs, cf [18, 20] The implementation of the new schemes may involve numerical differentiation However, we will show that the use of numerical differentiation does neither weaken the stability nor reduce the convergence order of the schemes The paper is organized as follows In the next section, we will review some nonstandard first order schemes proposed recently in [7] and explain why in certain cases they can be On BDF-Based Multistep Schemes for DAEs more efficient than the popular implicit Euler method Then, in Section 3, by extending the idea, we obtain a family of high-order difference schemes for a class of linear DAEs of index at most and for general linear DAEs of strangeness-free form These methods are proven to be convergent of the same order as that of the classical BDF methods applied to ODEs In Section 4, we discuss some other issues related with the implementations of the proposed schemes such as numerical differentiation and the effect of rounding errors In addition, the stability of the modified schemes is compared with that of the classical BDF methods as they are applied to the test DAE (7) and some numerical experiments are given for illustration Finally, in Section 5, we discuss the obtained results and suggest some possible future works First-Order Schemes As mentioned above, a number of standard implicit difference schemes applied to DAEs either produce unstable numerical solutions or require small step size In this section, we briefly review the investigation in [7] Let us define on the interval [0, 1] a uniform mesh consisting of meshpoints tj = j h, j = 0, 1, , M, h = 1/M and denote Aj = A(tj ), Aj = A (tj ), Bj = B(tj ), fj = f (tj ), xj ≈ x(tj ), x0 = x(0) Note that we may consider non-uniform meshes as well We consider two difference schemes that mimics the implicit Euler method Ai (xi+1 − xi ) + hBi+1 xi+1 = hfi+1 , i = 0, 1, , M − 1; (8) Ai+1 (xi+1 − xi ) + hBi xi+1 = hfi+1 , i = 0, 1, , M − (9) Applying scheme (8) to DAE (6) and omitting elementary computations, we obtain ui+1 = ψi+1 −αti+1 vi+1 , vi+1 = (ϕi+1 − ψi+1h−ψi )/(α−1) Comparing with the exact solution, we see that the numerical solution is convergent of first order to the exact solution If we apply scheme (9) to the same equation, then, it is easy to show that when α ∈ (−3; 1) we obtain an unstable process In this case the error grows at the rate of geometric progression 1+α Applying scheme (8) to DAE (7), we obtain ui+1 = (1 + αti+1 )vi+1 , vi+1 = vi , 1−z z = λh, which is unconditionally stable for any λ < and any α However, if we apply scheme (9) to the same equation, we obtain ui+1 = (1 + αti )vi+1 , vi+1 = − 2ω vi , − 2ω − z which may be unstable for certain values of z = λh and ω = αh It is clearly seen that, for these examples of DAEs (6) and (7), the scheme (8) is better than the classical Euler method and the scheme (9) M V Bulatov et al We will explain why scheme (8) in a number of cases is more efficient than the implicit Euler method and scheme (9) We rewrite the general linear DAE (1) in the form (A(t)x(t)) + (B(t) − A (t))x(t) = f (t), (10) and apply the implicit Euler method to the reformulated equation (10) Then, we have (Ai+1 xi+1 − Ai xi ) + h(Bi+1 − Ai+1 )xi+1 = (Ai+1 − hAi+1 )xi+1 − Ai xi + hBi+1 xi+1 = hfi+1 (11) Taking into account that Ai+1 − hAi+1 ≈ Ai , then substituting Ai for Ai+1 − hAi+1 in (11), we obtain exactly the scheme (8) This means that the efficient scheme (8) is nothing, but the result of the implicit Euler method as applied to the reformulated form (10) The convergence of scheme (8) for a class of linear DAEs of index (at most) two was proven in [7] A restrictive assumption is needed, namely the matrix function P (t) in (4) is assumed to be constant, i.e., P (t) ≡ P Unfortunately, it seems to be impossible to relax this condition We illustrate this fact with a very simple example of DAEs of index Example Consider the index-2 DAE 0 u v + u v = ψ , that has the unique solution v = ψ, u = −ψ Multiplying both sides of the equation by a variable matrix function P (t) = t ,we get the DAE u u = + t v v t If we apply scheme (8) to this DAE, we obtain a SLAE h hti+1 ti + h ui+1 vi+1 = ti ui vi ψ + (12) hψi+1 , whose coefficient matrix is singular for all i In the next section, we extend the scheme (8) to obtain higher order multistep schemes Higher Order Schemes In this section, we propose a family of linear k-step methods that are based on the popular backward differentiation formula The BDF methods are well known to be efficient stiff solvers [6, 14] For the IVP for ODEs of the form x = g(x(t), t), x(0) = x0 , t ∈ [0, 1], (13) a BDF method is defined by k ρj xi+1−j = hg(xi+1 , ti+1 ), i = k − 1, k, , M − 1, h = 1/M, (14) j =0 where ρj , j = 0, 1, , k, are the method parameters The paramaters ρj are determined by differentiating the interpolation polynomial for x at ti+1 The sets of parameters up to k = are available in many textbooks on numerical analysis of ODEs, e.g., see [1, 13, 14] On BDF-Based Multistep Schemes for DAEs When k > 6, it is known that the BDF methods for ODE (14) are unstable (in the sense of zero-stability) In order to apply (14), the starting values xi , ≤ i ≤ k − must be given First, consider the following two particular cases of (1) and (10) A12 (t) 0 u v + Is B12 (t) In−s u v = g(t) q(t) (15) , u Is B12 (t) u g(t) A12 (t) + = , (16) 0 v In−s v q(t) where u and g are s-dimensional, v and q are (n − s)-dimensional vector functions; A12 and B12 are smooth matrix functions of appropriate size We stress again that there is no constant-rank restriction on A12 These problems have the unique solutions v(t) = q(t), u(t) = g(t) − B12 q(t) − A12 q (t), and v(t) = q(t), u(t) = g(t) − B12 q(t) − (A12 q(t)) , respectively For numerical solution of problems (15) and (16), we apply difference schemes of the form k ζj xi+1−j + hBi+1 xi+1 = hfi+1 , i = k − 1, , M − 1, Ai+1 (17) j =0 where h k j =0 ζj xi+1−j is an approximation of order l ≥ to x (ti+1 ), i.e., h k ζj xi+1−j − x (ti+1 ) = O(hl ), j =0 provided that x is sufficiently smooth, and k ζj Ai+1−j xi+1−j + hBi+1 xi+1 = hfi+1 , i = k − 1, , M − 1, (18) j =0 where h1 ti+1 , i.e., k j =0 ζj Ai+1−j xi+1−j h is an approximation of order l ≥ to (A(t)x(t)) at t = k j =0 ζj Ai+1−j xi+1−j − (A(t)x(t))t=ti+1 = O(hl ), provided that A(t)x(t) is sufficiently smooth The following lemma is obtained for the convergence of schemes (17) and (18) Lemma Consider problems (15) and (16) together with discretization schemes (17) and (18) Let the following conditions hold The coefficients A12 , B12 and the right-hand side functions g, and q belong to class C l+1 [0, 1], l ≥ 1; The starting values x0 , x1 , , xk−1 are given and satisfy xj − x(tj ) = O(hl+1 ), ≤ j ≤ k − 1; k and h1 j =0 ζj xi+1−j (A(t)x(t)) t=ti+1 with order l, h k j =0 ζj Ai+1−j xi+1−j respectively approximate x (ti+1 ) and M V Bulatov et al Then, the difference schemes (17) and (18) are convergent to the exact solution of problem (15) and (16), respectively, with convergence order l, i.e., xi − x(ti ) = O(hl ), i = k, , M Proof Simple computations show that schemes (17) and (18) yield the formulae vi+1 = qi+1 , ui+1 = gi+1 − B12 (ti+1 )vi+1 − A12 (ti+1 ) h k ζj vi+1−j , j =0 and vi+1 = qi+1 , ui+1 = gi+1 − B12 (ti+1 )vi+1 − h k ζj A12 (ti+1−j )vi+1−j , j =0 respectively, for i = k − 1, k, , M − Thus, we obtain vi+1 = v(ti+1 ), i = k − 1, k, , M − 1, and the estimates ui+1 − u(ti+1 ) = O(hl ), i = 2k − 1, 2k, , M − 1, follow from the third condition of the lemma The estimates ui − u(ti ) = O(hl ) for i = k, k + 1, , 2k − 1, result from the first and the second conditions of the lemma The lemma has been proven We note that in the case of these special problems, a stability condition for the difference schemes (17) and (18) is not required; only the approximation is sufficient for the convergence Further, the smoothness of the coefficients as well as that of the solutions can be relaxed We give the following example for illustration Example Consider the DAE A(t)x (t) + x(t) = f (t), (19) where matrix A(t) has block form A(t) = A12 (t) 0 , and A(t) and f (t) are sufficiently smooth DAE (19) is of index and it has the unique solution x(t) = f (t) − A(t)f (t), which is independent of the initial data Applying to (19), the following first order difference schemes Ai+1 (xi − xi−1 )/ h + xi+1 = fi+1 , Ai+1 (−xi+1 + 3xi − 2xi−1 )/ h + xi+1 = fi+1 , which are known to be unstable for ODEs, we obtain fi − fi−1 xi+1 = fi+1 − Ai+1 h and −fi+1 + 3fi − 2fi−1 , xi+1 = fi+1 − Ai+1 h i = 1, 2, , M − 1, respectively It is clearly seen that both the numerical solutions are convergent of first order On BDF-Based Multistep Schemes for DAEs Based on this lemma, we are now in position to state the main result Reformulating (1) into the form (10), let us apply the k-step methods that are based on the backward differentiation formulae with k ≤ 6, we have the following difference scheme k ρj Ai+1−j xi+1−j + h(Bi+1 − Ai+1 )xi+1 = hfi+1 , i = k − 1, k, , M − (20) j =0 The following theorem shows the convergence of scheme (20) for a class of DAEs of the form (1) Theorem Consider the IVP (1)–(2) and suppose that the following conditions hold The coefficients A and B are real analytic functions and the right-hand side function f belongs to C k+1 [0, 1]; The DAE (1) has index at most and it can be reduced to canonical form (4) by a pair of nonsingular matrix functions P and Q, where P (t) = P is a constant matrix; The starting values satisfy x(tj ) − xj = O(hk+1 ), j = 0, 1, , k − Then, the scheme (20) with k ≤ is convergent of order k, i.e., the estimate xi − x(ti ) = O(hk ) holds for i = k, k + 1, , M and h = 1/M Further, the scheme (20) is as stable as its corresponding BDF applied to the underlying ODE Proof By virtue of the second condition of the theorem and formula (4), without loss of generality, we assume that 0a1 ×a1 N1 (t) N (t) = , 0a1 ×a2 0a2 ×a2 where 0ai ×aj , i, j = 1, are zero matrices of size × aj and N1 (t) is a matrix function of size a1 × a2 , d + a1 + a2 = n For simplicity, if no confusion arises, we omit the subscripts indicating the size of the blocks We multiply system (10) by matrix P and make the substitution x(t) = Q(t)y(t) Then, the transformed matrices P A(t)Q(t) and P B(t)Q(t) − P A (t)Q(t) = P B(t)Q(t) + P A(t)Q (t) − (P A(t)Q(t)) have the block form ⎛ ⎞ I 0 P A(t)Q(t) = ⎝ 0 N1 (t) ⎠ , (21) 0 ⎛ ⎞ J (t) 0 (22) P B(t)Q(t) − P A (t)Q(t) = ⎝ I −N1 (t) ⎠ 0 I Using (21) and (22), we obtain the following DAE of block form ⎞⎤ ⎞⎛ ⎡⎛ u(t) I 0 ⎣⎝ 0 N1 (t) ⎠ ⎝ v(t) ⎠⎦ + w(t) 0 ⎛ ⎞ ⎞ ⎛ ⎞⎛ J (t) 0 ϕ(t) u(t) ⎝ I −N (t) ⎠ ⎝ v(t) ⎠ = ⎝ ψ(t) ⎠ , ξ(t) w(t) 0 I (23) where (uT (t), v T (t), w T (t))T = y(t) and (ϕ T (t), ψ T (t), ξ T (t))T = Pf (t) In formula (20), we make the corresponding change of variable xi+1−j = Q(ti+1−j )yi+1−j , j = 0, 1, k and multiply the equation by P Due to formulae (21) and M V Bulatov et al (22), we obtain the following scheme in block form ⎛ ⎞ ⎞⎛ I 0 ui+1−j ρj ⎝ 0 N1,i+1−j ⎠ ⎝ vi+1−j ⎠ j =0 0 wi+1−j ⎛ ⎞⎛ ⎞ ⎞ ⎛ Ji+1 0 ui+1 ϕi+1 +h ⎝ I −N1,i+1 ⎠ ⎝ vi+1 ⎠ = h ⎝ ψi+1 ⎠ , wi+1 ξi+1 0 I k (24) T T T T where (uTi , viT , viT )T = yi = Q−1 i xi , (ϕi , ψi , ξi ) = Pf (ti ) We note that the difference system (24) is just the discretization of the differential system (23) by the BDF methods Due to the first and the third conditions of the theorem, the estimate ui −u(ti ) = O(hk ), i = k, k+1, , M is straightforward from the convergence of BDF methods for ODEs, (e.g., see [1, 6, 13]) The estimates vi − v(ti ) = O(hk ) and wi − w(ti ) = O(hk ), i = k, k + 1, , M follow from Lemma The (zero, absolute) stability of the scheme (20) is reduced to that of the corresponding BDF applied to the underlying ODE for component u The proof of the theorem is complete Remark It is clear that the standard canonical form (3)-(4) is not unique However, under the assumptions of Theorem 1, the dynamics (asymptotic behaviour, stability) of the underˆ are another pair of transformation matrices that lying ODE is invariant Indeed, if Pˆ and Q satisfies also the second condition of Theorem and the corresponding coefficient matrices are Nˆ and Jˆ, respectively, then by [2, Theorem 2.1], there exists a nonsingular constant matrix C1 such that Jˆ = C1 J C1−1 The latter relation implies that the underlying equations have the same stability property, i.e., they are kinematically equivalent We just showed that for the reformulated system (10), the transformation to the canonical form (21-22) and the discretization by the BDF methods commute This is the key to the convergence and stability analysis of scheme (20) Note that this commutativity of the transformation and the discretization does not hold for the original system (1) in general Remark We note that it is impossible to relax the condition P (t) = P in Theorem We consider again DAE (12) as an example Rewriting the equation in the form (10), we obtain t x + t x= ψ(t) If we apply methods (20) to this equation, then at each step of integration we obtain a singular system for xi+1 , since the coefficient matrix h ρ0 hti+1 ρ0 ti+1 is obviously singular When applying (20) to higher-index DAEs, the convergence result of Theorem does not hold The reason is that second-order or even higher-order derivatives of inhomogeneity are hiddenly involved which may cause the order reduction in numerical solutions On BDF-Based Multistep Schemes for DAEs For linear DAEs of index at most 1, the first and the second assumptions on the coefficients in Theorem can be relaxed We suppose that the DAE (1) is of the so-called strangeness-free form, see [17, pp 74, 93] and [18, 20], i.e., A(t) = A1 (t) B1 (t) B2 (t) , B(t) = (25) , where A1 , B1 are d by n matrix functions and B2 is a (n − d) by n matrix function such that A1 (t) B2 (t) is nonsingular for allt ∈ [0, 1] (26) For example, the test DAE (7) is strangeness-free It has been shown that under certain hypotheses, a general linear DAE of the form (1) can be reduced to the strangeness-free form, see [17] For sufficiently smooth A, B,and f , the DAE (1) with coefficients satisfying (25),(26) has differentiation index at most Lemma Consider DAE (1) with coefficients satisfying (25), (26) Assume further that A is at least continuously differentiable, B and f are continuous Then, there exists a continuously differentiable pointwise-nonsingular n by n matrix function Q such that by the change of variable x = Qy, the DAE (1) is transformed into the semi-explicit form ˜ ˜ + B(t)y = f (t), that is A(t)y ˜ = A(t) Id 0 ˜ , B(t) = B˜ 11 (t) B˜ 12 (t) B˜ 21 (t) B˜ 22 (t) (27) Proof First, since A1 is full row-rank and continuously differentiable, there exists a pointwise nonsingular and continuously differentiable matrix function Q such that AQ = A11 0 =: A, where A11 is a nonsingular and continuously differentiable d by d matrix function We set B = BQ + AQ Next, we define ˆ = Q −1 A11 0 ˆ the change of variable x = Qy transforms (1) into the form (27), Then, with Q = QQ, ˆ + AQ ˆ where B˜ = B Q Now, by the analogous arguments as those in the proof of Theorem 1, we obtain the following theorem for the scheme (20) when it is applied to DAEs of form (25) Theorem Consider the DAE (1) with initial condition (2) and suppose that the following conditions hold The coefficients A(t), B(t), and f (t) are such that A ∈ C k+1 [0, 1] and B, f ∈ C k [0, 1]; The DAE (1) is strangeness-free, i.e., (25) and (26) hold true; The starting values satisfy x(tj ) − xj = O(hk ), j = 0, 1, , k − 1 Then, the scheme (20) with k ≤ is convergent of order k, i.e., the estimate xi − x(ti = O(hk ) holds for i = k, k + 1, , M and h = 1/M Further, the scheme (20) is as stable as its corresponding BDF applied to the underlying ODE M V Bulatov et al Table Actual errors of the numerical solution by the 2-step BDF-based method (28) for the IVP (7) with α = 3, λ = −2 h 0.2 0.1 0.05 0.025 0.0125 eru 5.3795e-02 1.4343e-02 3.6059e-03 9.0210e-04 2.2555e-04 erv 1.3449e-02 3.5858e-03 9.0148e-04 2.2552e-04 5.6388e-05 Remark Since strangeness-free DAEs are of index at most 1, Theorem can be considered as a special case of [15, Proposition 3] However, here our proof avoids the use of projectors and therefore, it is less complicated than that of [15, Proposition 3] We note also that in [18], another approach is suggested for improving the stability of numerical methods for solving DAEs (25), but it requires more computational effort Further Issues and Numerical Experiments Now, we discuss the implementation of the scheme (20) In order to realize (20), the derivative A (t) is required at the mesh points If the exact values are not available, A (t) can be approximated by using the same backward differentiation formulae In this case, we obtain the modified scheme k ρj Ai+1−j xi+1−j + Di+1 xi+1 = hfi+1 , i = k − 1, k, , M − 1, (28) j =0 where Di+1 is calculated using the rule Di+1 = hBi+1 − to show that the convergence order for (28) is preserved k j =0 ρj Ai+1−j It is not difficult Theorem Consider the IVP (1)–(2) and let the same assumptions hold as in Theorem or Theorem Then, the scheme (28) with k ≤ is convergent of order k, i.e., the estimate xi − x(ti = O(hk ) holds for i = k, k + 1, , M and h = 1/M Further, the scheme (20) is as stable as its corresponding BDF applied to the underlying ODE with coefficients subject to perturbations of magnitude O(hk ) Proof First, we consider the IVP (1)-(2) under the assumptions of Theorem Due to the assumption on the approximation order of the numerical differentiation scheme, we have k ρj Ai+1−j − hAi+1 = O(hk+1 ), j =0 which implies ⎛ P⎝ k ⎞ ρj Ai+1−j − hAi+1 ⎠ Qi+1 = O(hk+1 ) (29) j =0 Due to the structure of P A(t)Q(t) in (21), the last block row of P A(t) is zero Hence, the last row block of P kj =0 ρj Ai+1−j Qi+1 is zero, too On the other hand, P Ai+1 Qi+1 itself does not depend on h Consequently, the last block row of the left hand side of (29) must be identically zero On BDF-Based Multistep Schemes for DAEs Now, we again multiply both sides of (28) by P from left and make the variable change xi+1−j = Qi+1−j yi+1−j Due to the above observation and taking into account the discretized system (24), we arrive at a structuredly perturbed system ⎛ ⎞ ⎞⎛ I 0 ui+1−j k ρj ⎝ 0 N1,i+1−j ⎠ ⎝ vi+1−j ⎠ j =0 0 wi+1−j ⎞⎛ ⎞ ⎞ ⎛ ⎛ k k k Ji+1 + O(h ) O(h ) O(h ) ui+1 ϕi+1 O(hk ) I + O(hk ) −N1,i+1 + O(hk ) ⎠ ⎝ vi+1 ⎠ = h ⎝ ψi+1 ⎠ , (30) +h ⎝ wi+1 ξi+1 0 I The last equation of (30) immediately yields wi+1 = ξi+1 , i.e wi+1 − w(ti+1 ) = for i = k − 1, , M − Now, consider the first and the second equations, which read k k k+1 )v k+1 )w i+1 + O(h i+1 j =0 ρj ui+1−j + h(Jj +1 + O(h ))ui+1 + O(h k k+1 )u k ))v N w + O(h + h(I + O(h + i+1 i+1 j =0 1,i+1−j i+1−j +h(−N1,i+1 + O(hk ))wi+1 = hψi+1 = hϕi+1 , (31) It follows that vi+1 = (−N1,i+1 wi+1 + ψi+1 )(I + O(hk )) + O(hk )ui+1 + O(hk ) Substituting this into the first equation of (31), we obtain k ρj ui+1−j + h Ji+1 + O(hk ) ui+1 = h ϕi+1 + O(hk ) j =0 This is nothing but the discretization of the underlying ODE u + J (t)u = ϕ(t) with perturbations O(hk ) by the corresponding BDF method By the standard argument of the convergence analysis for BDF methods [1, 6, 13], one can verify that ui+1 − u(ti+1 ) = O(hk ) for i = k − 1, , M − As a consequence, we also have vi+1 − v(ti+1 ) = O(hk ) for i = k − 1, , M − The convergence analysis of (28) for the IVP (1)-(2) under the assumptions of Theorem is similar Remark When implementing schemes (20) and (28), rounding errors arise as well The effect of rounding errors can be interpreted as follows We assume that, instead of the exact values fi+1 , we compute the numerical solution from (20) or (28) with perturbed values fˆi+1 satisfying h fˆi+1 − fi+1 ≤ ε, where ε is a rounding-error bound Thus, computational errors of magnitude O(ε) must be added to the right-hand sides of (29) and (30) Thanks to the transformation to the standard canonical form, the effect of rounding errors Table Actual errors of the numerical solution by the 3-step BDF-based method (28) for the IVP (7) with α = 3, λ = −2 h 0.2 0.1 0.05 0.025 0.0125 eru 1.4109e-02 2.0813e-03 2.6737e-04 3.3690e-05 4.2221e-06 erv 3.5271e-03 5.2033e-04 6.6843e-05 8.4226e-06 1.0555e-06 M V Bulatov et al Table Actual errors of the numerical solution by the 2-step BDF-based method (28) for the IVP (7) with α = 30, λ = −20 h 0.2 0.1 0.05 0.025 0.0125 eru 4.4960e-02 3.6646e-04 3.2789e-06 6.3627e-08 2.6566e-08 erv 1.4503e-03 1.1821e-05 1.0577e-07 2.0525e-09 8.5697e-10 can be analyzed without difficulty in the same manner as the proof of Theorem One can conclude that the accumulated error of the computed numerical solution by scheme (28) is of magnitude O(hk ) + O(ε/ h) Remark To initialize the schemes (20) and (28), k initial values xi , i = 0, 1, , k − 1, are needed As in the case of linear multistep methods for ODEs, the initial values xi are computed recursively by the methods of the same family Moreover, note that it is possible to extend the schemes (20) and (28) to a non-uniform mesh, i.e., the stepsize h may be variable, as done for ODEs, see [1, 6, 13] Next, we compare the classical BDF methods with the methods (20) as they are applied to the test DAE (7) Applying the classical BDF methods, one has k ρj xi+1−j + hBi+1 xi+1 = hfi+1 , i = k − 1, k + 1, , M − Ai+1 j =0 Decomposing xi = (uTi , viT )T and omitting elementary calculations, we obtain ui+1 = (1 + αti+1 )vi+1 , and find vi+1 from the recurrence relation (ρ0 − z − ω)vi+1 + (ρ1 − ω)vi + · · · + (ρk − kω)vi+1−k = 0, where z = hλ, ω = hα The coefficients of the latter equation depend on both λ and α However, the exact solution v(t) = exp(λt) does not depend on α Obviously, we can always find ω such that this scheme is unstable Then we apply scheme (20) to the same problem (7) Again omitting simple calculations, we obtain k ui+1 = (1 + αti+1 )vi+1 , ρj vi+1−j = zvi+1 j =0 It is clearly seen that the formula for component v is exactly the discretization of equation v = λv by the efficient BDF methods, which is independent of ω Finally, we carry out several numerical experiments to illustrate the efficiency of schemes (20) and (28) Table Actual errors of the numerical solution by the 3-step BDF-based method (28) for the IVP (7) with α = 30, λ = −20 h 0.2 0.1 0.05 0.025 0.0125 eru 2.7886e-02 1.3033e-02 1.0304e-05 9.2822e-08 6.7769e-09 erv 8.9955e-04 4.2043e-04 3.3238e-07 2.9943e-09 2.1861e-10 On BDF-Based Multistep Schemes for DAEs Example Numerical experiments with the k-step BDF-based methods (28), k = 2, 3, are carried out for the IVP (7) on the interval [0, 1] We use the uniform mesh with different values of stepsize h The actual errors eru = |u(1) − uM | and erv = |v(1) − vM | are displayed in Tables 1, 2, 3, and for two different sets of parameters α and λ For the IVP (7) with α = 3, λ = −2, the second- and third-order convergences of the numerical solutions are confirmed, see Table and Table 2, respectively For the IVP (7) with α = 30, λ = −20, which is a stiff problem, the BDF-based methods (28) still produce stable numerical solutions, see Table and Table Example We have also carried out numerical experiments with the BDF-based methods (20) and (28) for the index-2 DAE (6), and the numerical results again confirm the convergence order stated in Theorems and Conclusion In this paper, we have proposed numerical discretization schemes for some classes of linear DAEs of index at most Our methods are based on the well-known BDF methods applied to the reformulated form of the DAEs Under the assumption that the coefficients can be tranformed into the standard canonical form, these BDF-based methods for DAEs are shown to preserve the convergence order and the stability property as for ODEs The classes of DAEs investigated in this paper include linear DAEs with variable-rank leading terms whose index may be switching between and For such DAEs, up to our knowledge, the only result was obtained in [10] We stress again that the results for index-2 DAEs obtained here are different from that in [10] since we not require the properly stated form The assumption for the existence of a constant transformation matrix P is certainly a limitation, but as we have seen, it cannot be omitted We note that for any DAEs that can be transformed into the standard canonical form, the matrix P can be made constant by an appropriate scaling The latter procedure is computationally expensive, but at least it is always realizable in the general case 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(28 ) for the IVP (7) with α = 30, λ = ? ?20 h 0 .2 0.1 0.05 0. 025 0.0 125 eru 2. 7886e- 02 1.3033e- 02 1.0304e-05 9 .28 22e-08 6.7769e-09 erv 8.9955e-04 4 .20 43e-04 3. 323 8e-07 2. 9943e-09 2. 1861e-10 On BDF-Based. .. results again confirm the convergence order stated in Theorems and Conclusion In this paper, we have proposed numerical discretization schemes for some classes of linear DAEs of index at most Our methods... preserving integration of index- 1 DAEs Appl Num Math 45, 175? ?20 0 (20 03) 16 Higueras, I., Măarz, R., Tischendorf, C.: Stability preserving integration of index- 2 DAEs Appl Num Math 45, 20 1? ?22 9 (20 03) 17