CC BY
17
MSC 34A09, 65N99 DOI: 10.14529/mmp170201
ON THE CONCEPT OF INDEX FOR PARTIAL DIFFERENTIAL
ALGEBRAIC EQUATIONS ARISING IN MODELLING OF PROCESSES
IN POWER PLANTS
V.F. Chistyakov, E.V. Chistyakova
Institute for System Dynamics and Control Theory SB EAS, Irkutsk,
Russian Federation
E-mail: [email protected], [email protected]
This paper addresses some classes of linear and quasi-linear partial differential algebraic equations (PDAEs), i.e. systems of partial differential equations with singular matrices multiplying the higher derivatives of the desired vector-function. Such systems do not belong to the class of the Cauchy - Kovalevskaya equations, and therefore do not not comply with known existence theorems. The current research focuses on the first order evolutionary systems with one variable and investigates PDAEs depending on the parameter. The concept of index for PDAEs is introduced and various statements of initial boundary problems are considered. The results obtained are used to simulate and analyze the heat and mass exchange processes in power plants.
Keywords: partial differential algebraic equation; partial derivatives; integral differential equations; solution space; index.
Introduction and Statement of the Problem
Consider an evolutionary system of partial differential equations
p
A(Dt,Dx)u := A(x,t)Dtu + Bj(x,t)D{u + C(x,t)u = f (x,t), (x,t) e U, (1)
j=i
where A(x,t), Bj(x,t), C(x,t) are (n x n)-matrices, U = X x T C R2, X = [xo,xi], T = [t0,t1], Dt = d/dt, Dx = d/dx, f (x,t), u = u(x,t) are the given and the desired vector-functions, respectively. It is assumed that
det A(x,t) = 0, det Bp(x,t) = 0 y(x,t) e U, (2)
and that the entries of (1) are sufficiently smooth in some domain U that includes U . The solution u(x, t) is searched for in the domain U. In this paper, we focus only on classic solutions.
In what follows, by the solution of (1) we understand any vector-function u(x,t) that has continuous partial derivatives in U with respect to x, t and turns (1) into an identical U
The statement of the problem for partial differential equations usually includes initial and boundary conditions. Here we consider the simplest cases:
Uj(x,t0) = 4>j(x), u(x0,t) = ^(t), Uj(x,t) = DJxu(x,t), j = 0,p, DXu(x,t) = u(x,t). (3)
Ever since the second half of the 20th century the field of mathematics addressing equations with a noninvertible operator at the evolutionary term has played an important role in various applications such as hydrodynamics (the Navier-Stokes equations), gas dynamics (the Euler equations), electric and thermal engineering [1-8].
The study of such equations began with the work by L.S. Sobolev [1], that is why they are often referred to as Sobolev equations [2]. It is quite common to treat such equations by transition to the differential equations in the Banach spaces
Av(t) + Bv(t) = f(t), t e T, (4)
where A, B are some operators that put (4) into correspondence to (1) in the Banach spaces, ker A = 0; and v(t), f(t) are the desired and the given vector-functions, correspondingly.
A significant contribution into this field of mathematics has been made by G.A. Sviridyuk and his followers (see, for example, [2-7] and the references listed there). Interesting results are also presented in [9-14]. Another approach to solving Sobolev equations suggests transition to singular in some sense partial differential equations with subsequent application of powerful methods of functional analysis [15, 16]. Some promising results have been obtained for systems (1) with constant coefficient matrices by employment of Furrier transformations and similar methods (see, for example, the fundamental monographs [17,18] and the references listed there).
Finally, during the last 15-20 years it has become popular to employ the approach based on the methods developed for the DAE theory [19-25]. According to the American Mathematical Society, the term DAE is used for systems of ordinary differential equations with a singular matrix multiplying the higher derivative of the desired vector-function. Index is a notion used in the theory of DAEs for measuring the distance from a DAE to its related ODE. The index is a nonnegative integer number that provides useful information about the mathematical structure and potential complications in the analysis and the numerical solution of the DAE. It also identifies the number of derivatives on which the solution to the given DAE depends. However, there is still no agreement on how to calculate the index of partial differential algebraic equations (PDAEs), and the current research aims to provide some clarity on this matter. We will address a special case of (1) that comprises partial differential equations, ordinary differential equations, and algebraic equations.
When studying PDAEs, we face the question whether we can classify them as hyperbolic, elliptic, or parabolic, because the classic theory of partial differential equations states that the type of the system predetermines the method of solution (see, for example, [26]). Therefore, in what follows, we say that a PDAE is hyperbolic if it can be split into: 1) a classic hyperbolic system; 2) differential subsystems with respect to x,t, where the second variable is treated as a parameter; 3) a subsystem with a unique solution, in particular, an algebraic system.
tx
omitted, if this does not lead to misunderstanding. The inclusion V(x,t) e Cij(U), i,j> 1, where V(x, t) is some matrix (in particular, a vector-function), denotes that all elements of V(x,t) have continuous partial derivatives up to orders i,j in the domain U. If i = j, then we say that the matrix V(x,t) is i times differentiate in the domain U. Vi(x) e Cl(X), V2(t) e Cl(T) denote i-times differentiate matrices V1 (x), V2(t). The
continuous matrices are denoted as V(x,t) E C(U), V1(x) E C(X), V2(t) E C(T) and r[V(x,t)] = maxjrank V(x,t), (x,t) E U}.
Now consider an example to illustrate some properties specific to PDAEs.
Example 1.
(1 ai «2N 0 0 0 | Dtu+ 0 0 0
+
0 0
.0
0
ext 0
DX u +
Here u = {u1 u2 u3)T , f = (/1 f2 f3) ' , 5, ai, i = 1,8 are numeric parameters, Y = j(x, t) is some smooth function, T stands for transposition. However, in this situation, if 5 = 1 and y = 0, the system is solvable for any f1 E C1'1(U), f2 E C1'1(U), /3 E C3'1(U),y E C^U), if g(x,t) = [y(x,t) — (t + t2)ext] = 0 y(x,t) E U. Indeed, the third equation of the system yields u3 = f3 — extu2. Substitute u3 into the second equation. We obtain u2 = (f2 — Dxf3 — Dlf3)/g(x,t). Therefore, the components u2, u3 are uniquely defined in the domain U and belong to C1'1(U^. Then, by substituting u2, u3 into the first equation, we obtain an equation of the hyperbolic type
Dtu1 + a3Dxu1 + a6u1 = x,t)f1 — a3u2 — a4 u3,
where x,t) = f1 — a1Dtu2 — a2Dtu3 — a4Dxu3 — a5Dxu3 — a7u2 — a8u3. Hence, we can say that the system is implicitly hyperbolic and the following equality is valid
a3 0 0
e
a4
xt + 2t 0
a5 1 0
Dxu +
a6 0 0
a7
1 xt
as 0 5
u
f.
т
щ = ф(х — a\(t — t0)) + ехр(а3в)ф(х — a3(t — s), s)ds,
to
where ф(г) is an arbitrary function.
Summarizing what has been said, we are drawn to the following conclusions: 1) the components u2, u3 are fixed functions. Hence, we can set initial and boundary conditions only in the form of the functions u2(x,t0), u2(x0,t), u3(x,t0), u3(x0,t);
u1
and boundary conditions u1(x0,t),u1(x,t0) that satisfy the consistency conditions at the point (xo, to). For exam pie, ^(x0) = ^(t0) etc. [26];
3) if we perturb the free term f3 = f3 + t sin(tx/t2)1 then it can be readily seen that at e ^ 0 the following relations are valid: \\f3 — f3||c(u) ^ 0, \\u2 — u2||c(u) ^ which means that the solution is highly sensitive to changes in the initial data.
t
1. Auxiliary Information
Definition 1. [27] A pseudo inverse of the (m x n)-matrix M(x,t), t E U is defined as an (n x m)-matrix M +(x,t) satisfying the following criteria
M(x, t)M +(x, t)M(x, t) = M(x, t), M +(x, t)M(x, t)M+(x, t) = M +(x, t),
(M + (x, t)M(x,t))T = M+(x, t)M(x,t), (M(x,t)M+(x,t))T = M(x, t)M+(x, t).
M+(x, t) exists for any matrix and any (x,t) E U. If the matrix M(x,t) is square and regular, then M-1(x,t) = M + (x,t), and M-1 (x,t) E Cij(U), if M(x,t) E Cij(U).
Lemma 1. Let M(x,t) E Ci,j(U) and rankM(x,t) = const = r W(x,t) E U. Then:
1. There exist square matrices L(x,t), R(x,t) E Cij(U) such that det L(x,t) = 0, det R(x,t) = 0 W(x,t) E U, L(x,t)M(x,t)R(x,t) = diag{Ir, 0}, where Iv is an identity matrix of dimension v;
2. There exists the matrix M +(x,t) E Cij(U).
If rank M(x,t) = const, (x,t) E U, then at least one element of M +(x,t) has a
U
the monograph [28].
Now consider a higher order DAE depending on a parameter
k
(Dt)u :=Y, Aj(x,t)Dju = f (x,t), (5)
j=0
k
Ak (Dx)u Aj (x,t)DX u = f (x,t), (6)
j=0
where (x,t) E U, Aj(x,t) are (n x n)-matrices at least from C(U), det Ak(x,t) = 0, the xt
Definition 2. The operator Qi(Dt) := Ylj=0 Lj(x,t)Dj, where Lj(x,t) are (nxn)-matrices from C(U), with the property
k
Qi(Dt) ◦ Ak(Dt)y = £ Aj(x,t)Djy Wy E Ck+1(U), det Ak(x,t) = 0 W(x,t) E U, j=0
is called the Left Regularizing Operator (LRO) for the DAE (5). The smallest possible number l is said to be the index of (5).
A similar definition of the LEO can be formulated for (6) by replacing Dt with Dx.
Lemma 2. If system (5) has index l, then the following alternative holds: det Ak(x,t) = 0 W(x,t) E U for l = 0, or det Ak(x,t) = 0, (x,t) E U for l> 0.
Proof Indeed, if l = 0, then L0Ak = Ak W(x,t) E U, where det Ak = 0 W(x,t) E U. However, if l > 0, then it follows from the definition of index that LlAk = 0 W(x,t) E U. This is valid for continuous matrices Ll and Ak if and only if det Ll = det Ak = 0 W(x, t) E U
□
In other words, if we assume that the LEO exists, then the condition det Ak(x,t) = 0, (x, t) E U x
Theorem 1. Let system (5) satisfy the conditions:
1. Aj(x,t) E Cm'i(U), j = 0,k, m = max{(k - l)n + r + 1, 21}, r = r[Ak(x,t)], i > 0, f (x,t) E Cl'i(U);
2. The system has the LRO in U, which coefficients are either continuous or i times
x
Then, system (5) is solvable for any f (x,t), and its solution for any fixed x E X can be written in the form
u(x,t)= Xd(x,t)c(x) + W (Dt)f (x,t),
(7)
t l-k
W (Dt)f (x,t) = K (x,t,s)f (x,s)ds + £ Cj (x, t)Dj f, t j=0
where Xd(x,t) is an (n x d(x))-matrix, K(x,t,s), Cj(x,t) are (n x n)-matrices smooth with respect to t, j = 0,l — 1 rank Xd(x,t) = d(x) Wt E T, c(x) is an arbitrary function. Ifc(x) E Ci(X),then u(x,t) E Cfc'¿(U).
t
If l < k, the vector-function in (7) has the form W(Dt)f (x,t) = J K(x,t,s)f (x,s)ds.
to
Proof. Denote ( = DtuT ... D(tk-1'luT^ . Then we can put the following first order
DAE into correspondence to (5):
( 0 Akx t)) DtZ + U t) am) Z = (f x t)) ' (x't} E U (8)
where v = (k — l)n, A = (yA1 A2 ... Ak-1J. System (5) has the LEO of the form diag{Iv, Q¡(Dt)}, and the proof is based on application of the statement that was proved in [29] for the situation when k = l and the coefficient matrices as well the free term depend t
(Dt) ◦ Ak(Dt)y = n¡(Dt)f(x,t), (x,t) E U. (9)
If in Definition 1 matrices Aj(x, t) and the vector-function Q¡ f (x,t) are either continuous or г-times differentiable with respect to x, then any solution y = y(x,t) to (9) is either continuous or г-times differentiable with respect to x. Hence, the solutions to (5) possess the same properties.
□
A similar theorem can be formulated for the DAE (6). Then, according to Theorem 1, the general solution to (6) can be written in the form of the equalities:
u = Xd(x, t)c(t) + W(Dx)f (x,t),
W(Dx)f (x,t)= í K(x,t,s)f (s,t)ds + ^ Cj(x,t)Di f.
j=0
xo
,, (10)
2. Index for Linear PDAEs
Now, using the results from the previous section, introduce the concept of index for PDAEs.
Definition 3. Let there exist an operator ^(Dt,Dx) :=Yllj=0 Lj(Dx)Dj, where Lj(Dx) = Ei=0 Li(x,t)Dlx, Li(x,t) are (n x n)-matrices from C(U), with the property
%(Dt,Dx) ◦ A(Dt,Dx)y = A(Dx)Dty + A(Dx)y Vy e Cl+1(U),
where
m mi
A(Dx) = Ai(x,t)Dix, A(Dx) = Mx,t)Dx,
i=0 i=0
Ai(x,t), Ai(x,t) are (n x n)-matrices from C(U); it is assumed that the operator A(Dx) has the LRO in the domain U. The smallest possible l is said to be the index of system
t
xt
x
t
initial data is sufficiently smooth, the original system can be reduced to a vector integral differential equation resolved with respect to the evolutionary term
Dtu + W(Dx)A(Dx)u = W(Dx)f (x, t) + Xd(x, t)c(t). (11)
Now suppose that the conditions of Lemma 1 are satisfied. Then, by multiplying system (9) by L(x, t) on the left and introducing the change of variable u = R(x,t)z, we obtain
( Ir 0 ) Dz + ( k]1(Dx) k?(Dx) ) z = 0 do)
V 0 o) DtZ +\Af(Dx) A?(DX)) z = g> W
where g = L(x,t)f, A\v (Dx) = Ep=1 Biv(x,t)Dx + Civ(x,t), Biv(x,t), Civ(x,t), = 1,2 are the blocks of the matrices LBpDp R, ••• jBj DJx R and LADtR +
L Ylj=1 BjD3xR + LCR, correspondingly.
Example 2. Set in Example 1 a1 = 0, a2 = 0. Then the system has the form of the relation (12), where
1UDx) = a3Dx + a6, A12(Dx) = (aD + a7 a5Dx + a8), Af (^x,
A11(Dx) = a3Dx + a6, a]2(Dx) = (aD + a7 a5Dx + a8), Af1 (Dx) = 0,
( f 0 ) D2 + ( ^ + » 1 ) Dx + ( x '0 ) •
t
( 1 0 ) D + ( ^(Dx) AlHDx) )
{0A?(Dx)) Dtu H 0 A fiDx)) U = f (13)
a 22/^ \ i xext 0 \ _2 ( xext + 2 0 \ _ f DtY(x,t) 0 , where Af2(Dx) = ( 0 0 1 D2+ ( 0 0 Dx+ ( 'xext 0 )• The operator
A22(Dx) is index 2, if g(x,t) = Y(x,t) - (t + t2)ext] = 0 V(x,t) e U. Here the LEO has
the form Q2(Dx) = D D' M°reover> in formula (10) we have that d = 0 and
-i
( ext 1\
W(Dx) = ( t) q ) &2(Dx). Therefore, system (11) in this case has the form
Du + ( ^ Wm?№m , ) « = (W f t ) f = (fA
t V Q W(Dx)A(Dx) ) \W(Dx)Dtf2J \f2j
Theorem 2. Let one of the following conditions be satisfied for system (12):
1. The operator k^2(Dx) has the LRO in the domain;
2. k22(Dx) = Q and the operator k\l(Dx) o kl2(Dx) has the LRO in the domain U.
Then: 1) under conditions of Theorem 1 the DAE (1) has index 1 with respect to t in the U
tU
Proof. Transform the DAE (1) to the form (12). Differentiate the second block equation
t
( 0 \ + (л 1 'Dx) л 12Dx)) z = n=( M g = (gЛ сi4)
V л21 (Dx) л22^х) ) DtzЧ Л21 (Dx) л22^х)) z = g= \Dtg2) , 9 = \g2) '
where k (Dx) = DtB2j(x,t)Dx + DtCj (x,t), j = 1, 2.
Multiply the first block equation of (12) by the operator k21(Dx) and deduct the result from the second equation. This yields a system A(Dx)Dtz + A(Dx)z = g, where the operators A(Dx) and ^¿(Dt, Dx) from Definition ) can be written in the form A(Dx) =
diag{Ir, k22(Dx)}, %(Dt,Dx)= (K-kf(^Dx) jQ^ DtL(x, t), Vt = di&g{Ir, DtIn-r}.
Let the second part of the statement be satisfied. If we again differentiate the second
t
k12(Dx), from the second one, we arrive at
( Ir M Dz J^Dx) л12(Dx) \z = h П5)
\0 0j Dtz +\HDx) -л2 1 тл 12(Dx) )z = h
where h = (gj Dgj - Л211(Dx)gJ)T , Ф^) = л21(Dx) - Л211(Dx)Л\1(Dx). Since the operator л^^^л^^^ has the LEO, we therefore fulfill the first part of the theorem and can perform similar transformations. Here operators from Definition 3 have the form A(Dx) = dia.g{I,., -Ki1(Dx)Л^;¿(Dx)}■.
(Dt,Dx) = -D) I J) -^Dx) i!) VM
□
Remark 2. The paper [23] graded systems (1), where the operator л22(Dx) has the LEO, as index (1,l)- According to this definition, the system from Example 1 has index (1,1) at 8 = 1 and index (1,2) at 8 = 1.
There is also one more important remark to be made. Let the first part of Theorem 2 be satisfied. Then system (12) entails the equality A^2(Dx)z2 = -A'l1(Dx)z1 + g2, where
(zJ zJ)T = z.
z1 z2
equations of the DAE (12). We arrive at the system of integral differential equations resolved with respect to the evolutionary term
Dtz1 + [A21(Dx) - A12W(Dx)Al1(Dx)]z1 = h(x,t), (16)
where h(x,t) = g1 - A]2(Dx)Xd(x,t)c(t) - Af(Dx)W (Dx )g2- Eq. (16) without its integral part is a linear differential equation, which can be further investigated to find out whether it belongs to the class of hyperbolic, parabolic, or elliptic equations.
A similar system can be derived if the second condition of Theorem 2 is satisfied.
3. Hyperbolicity Criteria for Singular Systems
In this section we discuss techniques for finding the index of system (1) as well as criteria for assigning the system to a certain type. It is quite challenging to actually construct the matrices L(x,t) and R(x,t) that would transform the original system to the
A(x, t)
A( t) = il - sin u(x,t) cos u(x,t) y cos u(x,t) 1 + sin u(x,t)
where g(x,t) is some arbitrary smooth function. It can be readily seen that this matrix
U
If system (1) is regular, i.e. det A(x, t) = 0 V(x, t) e U, the hyperbolicity is understood as in [26]. In what follows, we provide criteria which, in terms of input data, guarantee
tU
hyperbolic structure.
Definition 4. If the matrix pencil XA(x,t) + B(x,t) satisfies the conditions: 1• r[A(x,t)] = r; 2. det[XA(x,t) + B(x,t)] = ao(x,t)\r + ••• , ao(x,t) = 0 V(x,t) e U,
U
Definition 5. If the pencil of continuous matrices XA(x,t) + ¡iB(x,t) + C(x,t) satisfies the conditions:
1. r[A(x,t)]= r1 <n, r[(A(x,t)|B(x,t))] = r1 + r2 < n;
2. det[\A(x,t) + ¡iB(x, t) + C(x,t)] = ao(x,t)\riff2 + • • • , ao(x,t) = 0 V(x,t) e U,
U
in terms of [30], has a simple structure). Lemma 3. If:
1. A(x,t), B(x,t), C(x,t) e Cij(U);
,
2. The matrix pencil XA(x,t) + B(x,t) satisfies the rank-degree criterion in U;
3. The matrix pencil XA(x,t) + fB(x,t) + C(x,t) satisfies the double rank-degree
U
Then:
1. Compliance with the the second point of the lemma entails that there exists such square matrices P1(x,t), Q1x,t) e Chj(U) that det P1(x,t)Q1(x,t) = 0 V(x,t) e U,
P1 (x,t)[\A(x,t) + B(x,t)]Q1(x,t) = A^ + J(x0,t) 0j ; (17)
2. Compliance with the the third point of the lemma entails that there exists such square matrices P(x,t), Q(x,t) e Cij(U) that det P(x,t)Q(x,t) = 0 V(x,t) e U,
P (x,t)[AA(x,t) + fB(x, t) + C(x,t)]Q(x,t) =
/Ir 0 0\ (J(x,t) 0 B13(x,t)\ (Cn(x,t) C12(x, t) 0 \ = A I 0 0 0 I + f I 0 Ie 0 I + I C21(x,t) C22 (x, t) 0 I , (18) \0 0 0) \ 0 0 0 J \ 0 0 Iv)
where r +e +v = n, J(x,t), B13(x,t), Cij(x,t), i = 1,2 are the matrix blocks of corresponding dimensions.
Remark 3. If q = n - r (here this is equivalent to mnk(A(x,t)lB(x,t)) = n V(x,t) e U), then the simple structure condition coincides with the rank-degree criterion. When the matrices of the pencil A(x,t) + fB(x,t) + C(x,t) depend only on t and the matrices A(x,t), (A(x,t)lB(x,t)) have a constant rank in the domain, the lemma on reducibility to the form (18) was announced in [31]. The lemma for two variables was proved in [23]. In this work we omit the requirement for the ranks to be constant because they follow from condition 2 of Definition 5.
Theorem 3. Let in system (1):
1. A(x,t), Bp(x, t), p = 1, C(x,t) e Cij(U), i,j > 1;
2. The matrix pencil AA(x,t) + B1(x,t) satisfy the rank-degree criterion in U or matrix pencil AA(x,t) + fB1(x,t) + C(x,t) have a simple structure in U;
3. All roots of the polynomial
det[AA(x,t) + D(x,t)] = 0, (19)
where D(x,t) = B1(x,t) + [In - S(x,t)S+(x,t)]C(x,t), S(x,t) = (A(x,t) | B1(x,t)), are real and simple, and
5 < A1(x,t) < A2(x,t) < • • • < Ap(x,t), Ap+1(x,t) = 0, ••• , Ar (x,t) = 0 V(x,t) e U, where 5 is some real number.
Then:
1. System (1) has index 1 in the domain U with respeet to t;
2. System (1) is implicitly hyperbolic;
3. System (1) has index (1, 0) if matrix pencil AA(x,t) + B1(x,t) satisfy the rank-degree
U
4- System (1) has index (1, 0) if matrix pencil XA(x,t) + ßB1(x,t) + C(x,t) have a
U
Proof. Multiply (1) by the matrix P and introduce the change of variable u = Qz, where P, Q are matrices from Lemma 3. We obtain
I 0 0\ J 0 Bi3
0 0 0 1 Dtz + I 0 Ie 0 0 0 0 0 0 0
Dxz +
'
Uli G12 G13
G21 G22 G23 z = Pf,
(20)
0
0 Iv
where Gj(x, t), i = 1, 2, j = 1, 2, 3 Me blocks of the matrix PADtQ + PBDXQ + PCQ. It is readily seen that if we multiply the last line by G\3, G23 and deduct the result from
P
can initially be chosen so that these blocks are zero. Now prove that the eigenvalues of the matrix J coincide with the roots of the polynomial (19). Consider a polynomial
det Pdet[AA + D] det Q = det[XPAQ + PBQ + P(In - SQQ-1S+)P-1PCQ] =
det
A
'Ir 0 0 ^ 0 0 0 0 0 0,
+
J0
0
Bl3 0
h 00
+ Z
G11 G12 G21 G22 00
0 0 Iv
where Q
0 Z1 0 Ip
= diag{Q, Q}, Z = P(In — SQQ 1 S+)P \ Direct calculation shows that Z = where Z1 is some block, and det P det[AA + D] det Q = det PQ det[AIr + J].
P, Q J
diagonal. Rewrite system (20) as
I
0 Iq
00 00
P 0
0 0 0 0
\0 0 0 0j
Dtz +
J1 0 00
0 0
0 0 Ie \0 0 0
Bu\
B24
0 0
Dxz +
G11 G21
G31 0
G12 G22 G32 0
G13 G23 G33 0
0 0 0 Iv
z = Pf, (21)
where J1 = diag{A1, ••• ,Ap}. If we write down (21) as the DAE (12), then the corresponding blocks take the form
A11(Dx)
K(Dx)
J 9 •
Dx +
G11 G12 G21 G22
G11 G12 0 0
A12(Dx)
K(Dx)
0
B14
0
B24,
)d*+ (Z 3 ■
{'013 Dx+wG03 ).
where the operator Af2(DX) has an ^^O of the form diag{Ie, DxIv}. Taking into
J
□
x
4. The PDAEs Based Mathematical Models
In view of what has been said above, consider modelling of some processes in power plants. Such models include equations describing fluid motion (for instance, water, oil fuel etc.) in pipelines of the network. The motion of incompressible, viscous liquid substances is described by a system of the Navier - Stokes equations, which can be written down in the form of a PDAE
M0 ¡OCM. m
where u = (u1 u2 u3)T, g(Z,t) = (g1(Z,t) g2(Z,t) g3(Z,t) 0)T is a given vector-function. Uj = Uj (Z,t), j = 1, 3 are coordinate velocities of fluid particles at the point (Z,t) = (x,y,z,t), p = p(Z, t) is a pressure at the point (Z,t),
Au = DXщ + D2yu2 + D2zu3, div u = Dxщ + Dyu2 + Dzu3, grad u = (Dxщ Dyu2 Dzu3)T
are the Laplas operator, divergence, and gradient, correspondignly; w(u) is the Jacobian
u
A(Dt ,Dx,Dy ,DZ )U
M 0*90+(-dvi3 r -u=0■ (23»
The system is written in a dimensionless form, i.e. it is assumed that fluid viscosity and density are equal to 1. Various forms of systems (24), (23) have been studied in the immense number of research works. In particular, system (23) was considered on the basis of the transition to (4) (see, for example, [6], [12]). A number of Russian and foreign researchers tried to apply the DAEs theory to the investigation of (23), and the paper [32] seems to be the first work of such kind.
System (23) has the same structure as the DAE (12), so if we set Aj1 = AI3, A12 = -grad, A\ 1 = div, A22 = 0, then, by repeating the reasoning of Theorem 2 and taking into consideration divgrad = A, we obtain
*2 ° A(Dt,Dx.D,D)U = ( I3 —A ) Dt (p) + ( -Al JYgdad )(U) -
where
= ( —Y ?)(0- Dt (( -dlvlK 0 DJ - Y = d,vAI,
Using the expression D ^ = ^ Dt- wr^e down the operator
*z=( л sM -dJO * 40 0 dZ
Therefore, we can assume that system (23) has index 2 with respect to t.
Such a transformation is not possible if the DAE (24) is nonlinear. However, we can apply the operator
2 —
( I3 0 \ [( I3 0 \ + ( 00 \ V -div 1 y L\ 0 1 )
Dt
to the nonlinear system to reduce its index. As a result we get
Dt
(5 0)CM
—AI3 grad Y -A
\(u\ + ( w(u)u \ — f g(Z,t) \ (24) ) \p) + {—div^(u)u) — {—divg(Z,t)J ■ {M)
It is well-known from the DAE theory, that the index reduction considerably increases computational reliability.
The index is preserved whatever approximations we use. For example, expand the desired function and the known function in a Fourier series with respect to spatial variables
X
U(Z,t) — V Uv(t)e-i(v'Z\ g(Z,t) — V Gv(t)e-i(v'Z), i — V-1,
X
/ v
v=0
v=0
where v — (v1, v2, v3), Vj, j — 1, 3 are integer numbers,
Uv(t) — (uiv(t) U2v(t) U3v(t) Pv(t))T, Gv(t) — (giv(t) g2v(t) gsv(t) 0)T. Substitute the expansions obtained into (23). We derive an infinite sequence of the DAEs
d dt
1000 0100 0010 0000
Uv (t) +
( -qv 0 0 — i v1
0 —qv 0 — iV2
0 0 —qv — iV3
iv1 — i v2 —iV3 0
Uv (t) — Gv (t),
(25)
where qv — vi + + vSystem (25) has index 2 in terms of Definition 2, i.e. there exists the operator Q2 — L0 + L1(d/dt) + L2(d/dt)2, where L0, L1, L2 — (4 x 4) are matrices with constant elements. The operator can be constructed following the algorithm from Theorem 2.
The method of lines is another way of approximating partial differential equations. For the sake of simplicity, we consider a two-dimensional Navier - Stokes system in the domain U — G x [t0,t11, G — [0,1] x [0,1]. Introduce on G a uniform grid in the x and y directions with the time step h.
Consider the system
Vjk (t) — Avjk (t) + Ax Pj,k(t) — [vj,k (t)Ax Vjk (t) + Wjk (t)AxVj , k (t)] — fijk,
W jk k (t) — A Wjk k (t) + AyPj, k (t) — [vj, k (t)Ay Wj, k (t) + Wj, k (t)Ay Wj, k (t)] — f2,j,k,
AxVjk k (t) + Ay Wj k k (t) — 0, where the difference operators have the form:
AxSj,k — (9+!, k — Sj,k)/h, AxSj,k — (Sj,k — Sj-1,k)/h, AySj,k — (Sj,k+1 — Sj,k)/hj Ay^j,k — (Sj,k — Sj,k-1)/h, A — AxAy + AyAy,
(26)
j is an arbitrary grid function. The values of the desired functions are assumed to have been derived from the initial boundary conditions, if the index is zero. Introduce a vector-function
Un(t) = (Zi(t) Ш • • • Zn-i(t))T, Pn(t) = (Vl(t) V2(t) • • • щ-i(t))T,
where N = 1/h, Q(t) = (vj,i(t) Wj,i(t) vj,2(t) Wj,2(t) ••• Vj,N-i(t) wj,n-i(t)), Vj(t) = (pj,l(t) pj,2(t) • • • Pj,N-l(t)) and rewrite (26) as a DAE
where DN, MN, SN are the matrices of the appropriate dimension, FN(t) is a vector-function composed of the functions f1k, f2>j>k and components of the initial and boundary conditions.
To control where fluid flows as well as to control fluid pressure, it is common to use hydraulic circuits. The hydraulic circuit graph can be presented by a full (m x n)-matrix A of nodes and lines that identically describes the structure and the orientation of the circuit: aj = 1 if the line i comes from the node j; aj = — 1 if the line i comes into the node j aij = 0, if the node j does not belong to the line i (i = 1, n, j = 1, m). It is assumed that the first and the second Kirchhoff circuit laws are satisfied: 1) at any node the amount of fluid flowing into the node is equal to the amount of fluid flowing out of that node; 2) the sum of pressure drops in any closed loop is zero. The connection between the flow rate of the line i and the pressures pbx, i(t), pbix,i(t) on its ends is expressed as
where Ti(t) > 0 is an inertia parameter of the line, hi(t) is a hydraulic head, s0, i > 0 and Sii > 0 are pipe frictions corresponding to the stream-line and turbulent flows. The relations (28) and the equations following from the Kirchhoff laws can be written in the form of the DAE
where (Aj Aj) = AT, R = diag{ro(t),ri(t),... ,Vn(t)], So = diagjso д,5о , 2,-..,so ,n], So = diag{si,i j , 2) ■ ■ ■ j Si,n } \X(t)\ = {|xi(t)|j \x2(t)\,...j |xn(t)|}, X(t) is an n-dimensional vector-function of the flow rates in pipelines; P (t) is an m^dimensional vector-function of the unknown pressures at nodes; P*(t) is an m2-dimensional vector-function of the known pressures; m1 + m2 = m; H(t) is an n-dimensional vector-function of hydraulic heads; Q(t) is an m^-dimensional vector-function of inflows; rankA1 = m1. It was previously shown in [33] that if we have a non-linear term in the system, there exists an operator П2 = L0 + L1(d/dt) + L2(d/dt)2, where L0, L1, L2 are constant matrices of the appropriate dimension, that transforms the DAE (29) to the normal form. Due to the
(27)
Pbx,i(t) - Pbix,i(t) + hi(t) = Ti(t)xi(t) + So,iXi(t) + Si,i |xj(t)| Xi(t), (28)
(29)
fact that the block structure of (27) is identical to that one of (29), the same technique can be applied to prove existence of such an operator for the DAE (27).
Models of complex power plants are typically described by quasi-linear DAEs, the study of which is quite challenging even when we deal with the simplest models. Consider a quasi-linear PDAE that describes heat exchange in a steam straight-through boiler, which can be primitively represented as a pipe with flowing fluid (water, steam, vapor-water) heated by hot gases and emission from the fuel combustion.
The conservation laws allow us to write down the following PDAE
an 0 0\ ZuÄ /xv, 0 0 \ /uÄ 0 a22 0 | Dt ( U2 I + ( 0' 0 0 | Dx I u2 I + 0 0 0/ \u3j \0 0 xkj \u3J
( c1[t(u1,p) - u2] I i 0 I + ( -Ci[t(ui,p) - u2] + C2u2 - c3u I = ( q(x,t) I , (30)
\ -C2u2 + C3u J \ 0 J
where u1 is fluid heat content; u2 is pipe wall temperature; u3 is gas enthalpy; t(u1,p) is fluid temperature; p is fluid pressure; xv,, xK,g are the flow and gas rates in the lines v, k a11,a22, c1, c2 ,c2 are some parameters responsible for the circuit general geometry and properties of hear exchange; q(x, t) is radiation heat flow. The gas flow rates and pressures are found when solving (29).
Problem (30) can be generalized as follows
A(x, t)Dtu + B(x, t)Dxu + C(u, x, t) = f (x, t), (x, t) e U,
where C(u, x, t) is a given in Rra x U vector-function, and applied to investigation of more relevant models.
Acknowledgements. This work has been supported by the Russian Foundation for Basic Research, grants No. 15-01-03228, 16-51-540002.
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Received February 28, 2017
УДК 517.9 Б01: 10.14529/ттр170201
О ПОНЯТИИ ИНДЕКСА ДИФФЕРЕНЦИАЛЬНО-АЛГЕБРАИЧЕСКИХ УРАВНЕНИЙ В ЧАСТНЫХ ПРОИЗВОДНЫХ, ВОЗНИКАЮЩИХ ПРИ МОДЕЛИРОВАНИИ ПРОЦЕССОВ В ЭНЕРГЕТИЧЕСКИХ УСТАНОВКАХ
В.Ф. Чистяков, Е.В. Чистякова
Институт динамики систем и теории управления им. В.М. Матросова СО РАН, г. Иркутск
В статье рассматриваются некоторые классы линейных и квазилинейных дифференциально-алгебраических уравнений (ДАУ) в частных производных. Под
ДАУ в частных производных в работе понимаются системы с вырожденными во всей области определения матрицами при старших производных искомой вектор-функции. Они не являются системами типа Коши - Ковалевской, и утверждения о разрешимости в общем случае отсутствуют. Конкретным объектом изучения являются эволюционные системы с одной пространственной переменной. Проведены исследования ДАУ высокого порядка, зависящих от параметра. На этой основе введено понятие индекса ДАУ в частных производных. Рассмотрены постановки начально-краевых задач для ДАУ в частных производных. Полученные результаты применяются для анализа моделей процессов тепломассообмена в энергетических установках.
Ключевые слова: дифференциально-алгебраические уравнения; частные производные; интегро-дифференциальные уравнения; пространство решений; индекс; модели энергетических установок.
Литература
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Виктор Филимонович Чистяков, доктор физико-математических наук, главный научный сотрудник, Институт динамики систем и теории управления им. В.М. Мат-росова СО РАН (г. Иркутск, Российская Федерация), [email protected].
JCjJlGHâ Викторовна Чистякова, К^НДИДсХТ физико-математических наук, научный сотрудник, Институт динамики систем и теории управления им. В.М. Матросова СО РАН (г. Иркутск, Российская Федерация), [email protected].
Поступила в редакцию 28 февраля 2011 г.
