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...

DOI 10.1007/s40306-016-0171-2

On BDF-Based Multistep Schemes for Some Classes

of Linear Differential-Algebraic Equations of Index

at Most 2

Mikhail Valeryanovich Bulatov 1,2 · Vu Hoang Linh 3 ·

Liubov Stepanovna Solovarova 1

Received: 6 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 2 is proposed It is shown that if the popular backward differentiation formulas (BDFs) are applied to a refor-mulated 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 Fur-ther 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

1 Matrosov Institute of System Dynamics and Control Theory, Siberian Branch, Russian Academy

of Sciences, Lermontov St., 134, Irkutsk, Russia

2 Irkutsk National Research Technical University, Lermontov St., 83, Irkutsk, Russia

3 Faculty of Mathematics, Mechanics and Informatics, Vietnam National University, 334, Nguyen Trai Str., Thanh Xuan, Hanoi, Vietnam

1 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, t f ], (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) ≡ 0 Such systems are called linear differential-algebraic equations

(DAEs) Without loss of generality, we assume in addition that[t0, t f ] = [0, 1] For DAE

(1), we assign initial condition

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

ris 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)=



I d 0

0 N (t)



P (t)A(t)Q(t) + P (t)B(t)Q(t) =



J (t) 0

0 I n −d



where I d 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) ≡ 0 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 do 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) =



I d 0

0 N (t)

 

y

1(t)

y(t)

 +



J 0

0 I n −d

 

y1(t)

y2(t)



=



ϕ(t) ψ(t)



where (y1T (t), y2T (t)) T = y(t),



ϕ(t) ψ(t)



= P (t)f (t) Thus, y1is a solution of the so-called

1+ J (t)y1= ϕ(t), and y2is calculated using the following rule, see [4],

y2(t)=

r−1



i=0

( −1) i T i ψ(t),

where the effect of the operator T on a vector-function ψ is defined by

T ψ(t) = N(t) d

dt ψ(t), T

0ψ(t) = ψ(t).

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 1 ([9]) The DAE



1 t

0 0

 

u

v

 +



0 α

1 t

 

u v



=



ϕ(t) ψ(t)





t 1

−1 0



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

u i+1= ψ i+1− t i+1v i+1, v i+1= 1

α



ϕ i+1+ v i−ψ i+1− ψ i

h



,

where t i = ih, i = 0, 1, , M, h = 1/M, ψ i = ψ(t i ) , ϕ i = ϕ(t i ) , u i ≈ u(t i ) , v i ≈ v(t i ) Thus, when|α| < 1, we obtain an unstable numerical solution (the error grows at velocity

every step of integration If for this equation we apply a number of other implicit

Runge-Kutta 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 numer-ical solution of DAEs of index 1 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

Example 2 ([18]) The IVP



 

u

v



=



−1 1 + αt

 

u v



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 1 and it is called the test equation for DAEs which generalizes the classical test equation for ODEs y = λy When λ << −1, the problem (7) is a stiff





If we apply the implicit Euler method, we obtain

v i+1= R(z, ω)v i , u i+1= (1 + αt i+1)v i+1,

where t i = ih, i = 0, 1, , M, h = 1/M, ω = αh, z = λh, u i ≈ u(t i ) , v i ≈ v(t i ), and

R(z, ω)= 1− z − ω1− ω

The function R(z, ω) is called the stability function associated with the implicit Euler

method [18], which generalizes the classical stability function defined for Dahlquist’s test

equation x  = λx Evidently, this stability function depends on ω as well and for any z < 0

we can choose values of ω such that |R(z, ω)| > 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 differ-ence 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 2 When discretizing, we take into account the structure of the problem The idea is that, instead of discretizing directly the original problem, we will do that for a refor-mulated 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 reformu-lated 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

switch-ing between 1 and 2, but we do not consider DAEs in properly stated form In our approach, DAEs that are discretized do not necessarily satisfy the assumptions for the numerically qualified form defined in those references Consequently, the convergence results for

index-2 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 sug-gested 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

more efficient than the popular implicit Euler method Then, in Section3, by extending the idea, we obtain a family of high-order difference schemes for a class of linear DAEs of index at most 2 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 Section4, 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 Section5, we discuss the obtained results and suggest some possible future works

2 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

A

j = A(t

j ) , B j = B(t j ) , f j = f (t j ) , x j ≈ x(t j ) , x0 = x(0) Note that we may consider

non-uniform meshes as well

We consider two difference schemes that mimics the implicit Euler method

A i (x i+1− x i ) + hB i+1x i+1= hf i+1, i = 0, 1, , M − 1; (8)

A i+1(x i+1− x i ) + hB i x i+1= hf i+1, i = 0, 1, , M − 1. (9)

u i+1= ψ i+1−αt i+1v i+1, v i+1= (ϕ i+1−ψ i+1−ψ i

solu-tion, 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



 2

1+α.

Applying scheme (8) to DAE (7), we obtain

u i+1= (1 + αt i+1)v i+1, v i+1= 1

1− z v i ,

z = λh, which is unconditionally stable for any λ < 0 and any α However, if we apply

scheme (9) to the same equation, we obtain

u i+1= (1 + αt i )v i+1, v i+1= 1− 2ω

1− 2ω − z v i ,

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)

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

(A i+1x i+1− A i x i ) + h(B i+1− A

i+1)x i+1

= (A i+1− hAi+1)x i+1− A i x i + hB i+1x i+1= hf i+1. (11)

Taking into account that A i+1− hA

i+1 ≈ A i , then substituting A i for A i+1− hA

i+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 2

Example 3 Consider the index-2 DAE



0 1

0 0

 

u

v

 +



u v



=

 0

ψ



,

that has the unique solution v = ψ, u = −ψ.

Multiplying both sides of the equation by a variable matrix function P (t)=



1 0

t 1



,we

0 1

0 t

 

u

v

 +



1 0

t 1

 

u v



=

 0

ψ



If we apply scheme (8) to this DAE, we obtain a SLAE



ht i+1 t i + h

 

u i+1

v i+1



=



0 1

0 t i

 

u i

v i

 +

 0

hψ i+1



,

whose coefficient matrix is singular for all i.

In the next section, we extend the scheme (8) to obtain higher order multistep schemes

3 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

a BDF method is defined by

k



j=0

ρ j x i +1−j = hg(x i+1, t i+1), i = k − 1, k, , M − 1, h = 1/M, (14)

where ρ j , j = 0, 1, , k, are the method parameters The paramaters ρ j are determined

by differentiating the interpolation polynomial for xat t

i+1 The sets of parameters up to

k= 6 are available in many textbooks on numerical analysis of ODEs, e.g., see [1,13,14]

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 x i, 0≤ i ≤ k − 1 must be given.

First, consider the following two particular cases of (1) and (10)



0 A12(t)

 

u

v

 +



I s B12(t)

0 I n −s

 

u v



=



g(t) q(t)





0 A12(t)

 

u v

 +



I s B12(t)

0 I n −s

 

u v



=



g(t) q(t)



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) − B12q(t) − A12q(t), and

v(t) = q(t), u(t) = g(t) − B12q(t) − (A12q(t)), respectively

For numerical solution of problems (15) and (16), we apply difference schemes of the form

A i+1

k



j=0

ζ j x i +1−j + hB i+1x i+1= hf i+1, i = k − 1, , M − 1, (17) where 1hk

j=0ζ j x i +1−j is an approximation of order l ≥ 1 to x(t

i+1), i.e.,

h1

k



j=0

ζ j x i +1−j − x(t i+1)

=O(h l ),

provided that x is sufficiently smooth, and

k



j=0

ζ j A i +1−j x i +1−j + hB i+1x i+1= hf i+1, i = k − 1, , M − 1, (18) where h1k

j=0ζ j A i +1−j x i +1−j is an approximation of order l ≥ 1 to (A(t)x(t)) at t =

t i+1, i.e.,

h1

k



j=0

ζ j A i +1−j x i +1−j − (A(t)x(t))

t =t i+1

=O(h l ),

provided that A(t)x(t) is sufficiently smooth.

The following lemma is obtained for the convergence of schemes (17) and (18)

Lemma 1 Consider problems (15) and (16) together with discretization schemes (17) and

(18) Let the following conditions hold.

1 The coefficients A12, B12and the right-hand side functions g, and q belong to class

C l+1[0, 1], l ≥ 1;

2 The starting values x0, x1, , x k−1are given and satisfy

x j − x(t j ) =O(h l+1),0≤ j ≤ k − 1;

3 1hk

j=0ζ j x i +1−j and 1hk

j=0ζ j A i +1−j x i +1−j approximate x (t i+1) and (A(t)x(t))t =t+1with order l, respectively.

Then, the difference schemes (17) and (18) are convergent to the exact solution of

prob-lem (15) and (16), respectively, with convergence order l, i.e., x i − x(t i ) = O(h l ),

i = k, , M.

Proof Simple computations show that schemes (17) and (18) yield the formulae

v i+1= q i+1, u i+1= g i+1− B12(t i+1)v i+1− A12(t i+1)1

h

k



j=0

ζ j v i +1−j ,

and

v i+1= q i+1, u i+1= g i+1− B12(t i+1)v i+1− 1

h

k



j=0

ζ j A12(t i +1−j )v i +1−j ,

respectively, for i = k − 1, k, , M − 1 Thus, we obtain

v i+1= v(t i+1), i = k − 1, k, , M − 1,

and the estimates

u i+1− u(t i+1) = O(h l ), i = 2k − 1, 2k, , M − 1,

follow from the third condition of the lemma The estimates u i − u(t i ) = O(h l ) 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 conver-gence Further, the smoothness of the coefficients as well as that of the solutions can be relaxed We give the following example for illustration

Example 4 Consider the DAE

A(t)x(t) + x(t) = f (t), (19)

where matrix A(t) has block form

A(t)=



0 A12(t)



,

and A(t) and f (t) are sufficiently smooth DAE (19) is of index 2 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

A i+1(x i − x i−1)/ h + x i+1= f i+1,

A i+1( −x i+1+ 3x i − 2x i−1)/ h + x i+1= f i+1,

which are known to be unstable for ODEs, we obtain

x i+1= f i+1− A i+1f i − f i−1

h

and

x i+1= f i+1− A i+1−f i+1+ 3f i − 2f i−1

i = 1, 2, , M − 1, respectively It is clearly seen that both the numerical solutions are

convergent of first order

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



j=0

ρ j A i +1−j x i +1−j + h(B i+1− Ai+1)x i+1= hf i+1, i = k − 1, k, , M − 1 (20)

The following theorem shows the convergence of scheme (20) for a class of DAEs of the form (1)

Theorem 1 Consider the IVP (1)–(2) and suppose that the following conditions hold.

1 The coefficients A and B are real analytic functions and the right-hand side function f belongs to C k+1[0, 1];

2 The DAE (1) has index at most 2 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;

3 The starting values satisfy x(t j ) − x j =O(h k+1), j = 0, 1, , k − 1.

Then, the scheme (20) with k ≤ 6 is convergent of order k, i.e., the estimate x i − x(t i ) =

O(h k ) 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

N (t)=



0a1×a1 N1(t)

0a1×a2 0a2×a2



,

where 0a i ×a j , i, j = 1, 2 are zero matrices of size a i × a j 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

P A(t)Q(t) − (P A(t)Q(t))have the block form

P A(t)Q(t)=

⎛

⎝I 0 0 N0 10(t)

⎞

P B(t)Q(t) − P A(t)Q(t)=

⎛

⎝J (t)0 0I −N0

1(t)

⎞

Using (21) and (22), we obtain the following DAE of block form

⎡

⎣

⎛

⎝I 0 0 N0 10(t)

⎞

⎠

⎛

⎝u(t) v(t) w(t)

⎞

⎠

⎤

⎦

 +

⎛

⎝J (t)0 0I −N0

1(t)

⎞

⎠

⎛

⎝u(t) v(t) w(t)

⎞

⎠ =

⎛

⎝ψ(t) ϕ(t) ξ(t)

⎞

where (u T (t), v T (t), w T (t)) T = y(t) and (ϕ T (t), ψ T (t), ξ T (t)) T = Pf (t).

Q(t i +1−j )y i +1−j , j = 0, 1, k and multiply the equation by P Due to formulae (21) and

(22), we obtain the following scheme in block form

k



j=0ρ j

⎛

⎝I 0 0 N0 1,i0+1−j

⎞

⎠

⎛

⎝u v i i +1−j +1−j

w i +1−j

⎞

⎠

+h

⎛

⎝J i0+1 I0 −N0

1,i+1

⎞

⎠

⎛

⎝u v i i+1+1

w i+1

⎞

⎠ = h

⎛

⎝ψ ϕ i i+1+1

ξ i+1

⎞

where (u T i , v i T , v i T ) T = y i = Q−1i x i , (ϕ i T , ψ i T , ξ i T ) T = Pf (t i )

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 u i −u(t i ) = O(h k ) , i = k, k+1, , M is straightforward from the convergence

of BDF methods for ODEs, (e.g., see [1,6,13]) The estimates v i − v(t i ) = O(h k )and

w i − w(t i ) = O(h k ) , i = k, k + 1, , M follow from Lemma 1 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 1 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 under-lying ODE is invariant Indeed, if ˆP and ˆQare another pair of transformation matrices that satisfies also the second condition of Theorem 1 and the corresponding coefficient matri-ces are ˆNand ˆJ, respectively, then by [2, Theorem 2.1], there exists a nonsingular constant

matrix C1such that ˆJ = C1J C−1

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 canon-ical 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 2 We note that it is impossible to relax the condition P (t) = P in Theorem 1 We

consider again DAE (12) as an example Rewriting the equation in the form (10), we obtain



0 1

0 t



x

 +



1 0

t 0



x=

 0

ψ(t)



.

If we apply methods (20) to this equation, then at each step of integration we obtain a

singular system for x i+1, since the coefficient matrix



h ρ0

ht i+1 ρ0t i+1



is obviously singular

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