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Initial-boundary value problems for a discrete hydrodynamical model (model d) with velocity boundary conditions are globally well posed
Math/ Comput. Modelling, Vol. 12, No. 8, pp. 959-966, 1989 Printed in Great Britain. All rights reserved
0895-7177/89 $3.00 + 0.00 Copyright 0 1989 Maxwell Pergamon Macmillan plc
INITIAL-BOUNDARY VALUE PROBLEMS FOR A DISCRETE HYDRODYNAMICAL MODEL (MODEL D) WITH VELOCITY BOUNDARY CONDITIONS ARE GLOBALLY WELL POSED D. L. HICKSand K. L. KUTTLER Department
of Mathematical
Sciences, Michigan
Technological
(Received November 1987; accepted Communicated
University,
for publication
Houghton,
MI 49931, U.S.A.
November 1988)
by X. J. R. Avula
Abstract-A discrete model for the one-dimensional motion of a hydrodynamical material is globally well posed when the boundaries have time dependent velocities imposed. The proof is given for the case where the material law is a generalization of the ideal gas form with a viscosity of the Navier-Stokes form. This modelling approach suggests a simple solution to the problem of the deterministic modelling of turbulent motions.
INTRODUCTION The standard, continuum model for hydrodynamical motion we refer to as “Model C”. For many years (going back at least to Perrin in 1906 [l]) some researchers have th;ought that the derivation of Model C from the first principles of classical, Newtonian physics involves some questionable steps [l-3]. One problem is that the molecular nature of matter makes the modelling of a material as a mathematical continuum somewhat mysterious. The classical argument used in deriving Model C involves three stages; (i) assume the material is partitioned into chunks (a chunk of material is a finite set of molecules) “large enough to behave as a continuum in some sense”; (ii) apply first principles to produce equations of motion for the chunks; (iii) take a limit as the chunk size goes to zero to generate the differential equations of Model C. Paradox: how can a finite set of molecules simultaneously be “large enough to behave as a continuum” but have its size go to zero? Model C has been used for many years with seemingly useful results. This may be at least partly an illusion. That is, the discrete analogs of Model C used in numerical methods (finite difference, finite elements, etc.) have apparently been responsible to some extent for this perception of the successfulness of Model C. There are many open questions in the the&y of global well posedness of problems arising from Model C, especially when the initial-boundary value data and material laws are physically realistic. We wonder if there is a connection between these open questions and the questionable derivation of Model C. Global well posedness theory should form the foundation for the study of the continuum and discrete equations representing the models for the motions of materials such as liquids, gases, etc. Unfortunately global well posedness questions for Model C seem to be extremely difficult and good results are few [4]. In a recent paper [3], we presented a discrete model, which we call Model D, that avoids the dubious derivation described above. The Model D approach leads to a system of equations for the evolution of the average energy, momentum, and volume of chunks of the material. Model D is derived by applying the first principles of classical Newtonian physics to chunks or finite cells of the material. If we were to give the method of deriving Model D a name, we might call it a “finite cell method”. Although Model D is not derived by applying a finite difference or finite element method to Model C, it does resemble Model C with derivatives discretized, of course. Global well posedness results are important in various numerical analysis problems that arise in computational continuum dynamics. For example, the analysis of convergence and error estimates is much easier if global well posedness has been established. Well posedness is important from a deterministic modelling perspective because in this discipline a single solution should exist and depend continuously on the initial and boundary data. 959
960
D. L. HICKSand K. L. KUTTLER
In a recent article in the journal Zak [5] presented a nice discussion of the problem of producing deterministic mathematical models for turbulent motions. Quoting from Zak [5]: “As shown by physical and numerical experiments, a complete description of chaotic motions is impractical since such motions are sensitive to small errors in initial conditions, and therefore, are non-reproducible no matter how precisely the experiment is repeated. However, the most striking characteristic of chaotic motions is the stability of certain averaged quantities, such as the mean velocity field of a turbulent motion. Indeed, despite the fact that the velocity at each particular point of a turbulent flow is unstable and unpredictable, the mean velocity profiles are usually stable and reproducible. . . . In view of these ideas it is reasonable to express the original mathematical models in terms of stable and reproducible quantities of chaotic motions”. We suggest that the Model D approach provides a simpler solution to the problem of the deterministic mathematical modelling of turbulent motions. We prove that Model D provides a stable, deterministic model because that follows, of course, from the global well posedness. Although this paper presents results only for the one-dimensional version of Model D, these results can be extended to higher dimensions as will be shown in future papers. What has been established for Model D: in [3] we presented the derivation of the model and showed that a problem involving the response of a material interval with both ends fixed was globally well posed. In [6] we showed global well posedness for the stress boundary case. In this paper global well posedness is proved for the case where velocity boundary conditions are prescribed. This is done for a material law that generalizes the case when the equation of state is of the ideal gas form with viscosity of the Navier-Stokes form. The initial data is assumed to be physically acceptable. The details are developed here only for the case where one end is fixed, but the case with nonzero time-dependent velocities prescribed on both ends can be analyzed in a similar manner. To begin, Model D is presented and then the global existence, uniqueness, and continuity of the solution operator is established. MODEL
D WITH
VELOCITY
BOUNDARY
CONDITIONS
Let the material interval be [pO,~~1 and to it assign a partition 0 = PO< p, <. . . < p,. Let the cross sectional area of the one-dimensional interval [cl,, pi+ ,] is
(1)
material be A. Then the mass of the material
Massbj7~,+Il=A(~,+I -II,).
(2)
It is convenient to take A = 1. Let V, u, E denote specific volume, specific momentum, specific total energy. Vol’[p,, cl/+,] is the volume of [p,,, pj+ ,] at time r. The average specific volume of [p,,, .P~+,] at time t is given by v(t, Pj+~~2)=Vol’[(Pj~ /++~llMN~~~Pj+ll
(3)
and E(t, pj+ ,!2), u(t, pj+ ,,*) are similarly defined. Model D, derived in [3], is given by g
(t, P,+ I?) =
u(r9P~+ I) - u(t3PjLi) Pj+l
g
(t, P,+ Iid =
-PI
E(t, P,+ I>- Z(t9 P,) Pj+l-Pj
$ ft3
Pj+
12)=
w(t3Pi+ I) - w(tYPj> Pj+I-II,
(4)
Discrete hydrodynamical model
961
for o
‘ttP Pji> = t”ttv Pj+ 19)+ u(t9Pj- I/*))/2 ‘tt,
(8)
Pj) = tctt2 Pj+ IL?)+ c(t, Pj- l/2)1/2
(9)
w(t, PjFci) = t”ttY Pj+ 1;2)C(t2Pj- l/7_) + utt, PI- IQ)C(t, /$‘I+IQ))/2 for
I sgj G r - 1. The stress Z(t, p,+ ,,?) is given by
with the specific internal energy e given by e(tY l(j+l/*)
=
tE
-
u2/2Nt,
Pj+
(11)
I/Z).
In equation (10) P is the pressure and 1 >, 0 is the coefficient of viscosity. P is given by a generalization of the ideal gas law: P(e, V) = k(e)h( V)
where k and h are positive for e and V positive and
(12)
s
’ de
S’
h(V)dV=co=
(13)
Ok(e)'
0
Recall that the ideal gas law is P(e, V) = Te/V where r > 0 is a constant. For the ideal gas law k(e) = Te and h(V) = l/V’; note that equation (13) is satisfied by the ideal gas law. In the Navier-Stokes viscosity A = A,p where I, > 0 is a constant. Thus the material law considered here is a generalization of the case where the equation of state is of the ideal gas form and the viscosity is of the Navier-Stokes form. Lipschitz conditions are assumed on h, k and Iz in all arguments for 0 < 6 < min(e, V) G max(e, V) 6 C < cc and 1~1< K c 00; the Lipschitz conditions may depend on 6, C and K. Note that no convexity assumptions are made on h, k or 1. The unknown functions in equation (4) are V, U, E at (t, pj+ ,,2) for 0
PO)
=
Z(t,
(14)
A,2)
a(t, PO)= 0
(15)
w(r, &I) = 0
(16)
at the right boundary Z(t, /A) = Z(t, P,-
(17)
l/2>
u(t, P,) = u,(t)
(18)
W(GP,) = a(t, PL,)C(tvcl,).
(19)
These conditions represent the left end of the material as fixed while the right end experiences a time dependent velocity u,; assume u, is continuous. Letting (x0, . . * 3X,-I, x,, . . . 3&-1,X2,,
* * . , X3-l) =
(Vf., Et.9
h/2)9 Pl,2),
. * * 7 V(. . . . 3 Eta
9 /A3 A-
1,2), UC.
7 Pl,Z),
' . * 9 4'3
ru,- I/Z)>
I/Z))
the system of differential equations (4) for Model D with conditions (7)-(17) may be written as x’(t) = F(t, x(t)).
(20)
D. L. HICKS and K. L. KUTTLER
962
LetS(6,C,K)={xER3’:O<6bmin(x, ,..., x,_,,yZ ,,..., y,, _ , ) d C and max(lx,I, . , jxz, _ , /) < Kj where y*l+i= Note
y,,_,)
.
r.~,_l,+YZ,, . . . ,
X2+, - (x”+,)*P
that Y2r+t = e(t,
i.e. y is the specific internal Let
!A+ 1;21r
energy.
U = U(S(6,
C, K):O < min(6,
C. K) Q max(b, C, K) < co}.
U is called the set of &&ally acceptable vectors. Note that the V, U, E which are constructed from components of the vectors in U are such that V, e are positive and V, (~1, e are bounded. If the initial data are in U then they are said to be physically acceptable. Note that the only requirement on the initial data is that it be physically acceptable. Let x(0) = x0
(21)
and 5, = (V:,,, . . . , V:_ ,,I, u::,, . . , u;_ ,;2, E?;2, .
. , E;_ ,iz).
Suppose
with 6,, C,, & strictly positive and finite and 6, < C,. Thus the initial V and e are between So and C, and the initial u is between -K, and + K,. The assumptions on h, k, ,l imply that for x, y E S(6, C, K), there exists a Lipschitz constant L(6, C, K) such that IjFk XI - F(r, Y>II6 U&
C, K)\jx - YI(
and the continuity of C, implies that F is continuous in t. The boundary velocity assumptions are as follows. BVA(0): cr(.) is continuous on [0, T]. BVA(i): let VO~~[~~,p,] be the total volume of [po, p,] at time t = 0. Let Vol’[po, p,] be defined
(24
by
I VOl’[Po, Prl = vo~“[Po, i&l+ Then
s0
Gz> dz
(23)
< cc,.
(24)
for all t in [0, T], 0 < Vol, < vol’[~o) pr] < Vol,
BVA(ii):
let ErgO[p,, p,] be the total energy of [pO, p,] at time t = 0. Let Erg’[p,, FL,]be defined
by
I Wbo7 P,I = Ergok0TAI+ Then
s0
OAT P (z>clr 1dr.
(25)
for all t in [0, T] 0 < Erg, < Erg’[po, ~~1 Q Erg,
< 00.
(26)
In the proof of Lemma 2 it is shown that Vol’[p,, p,] is the total volume and Erg’[po, p,] is the total energy at time t of [po, IL,]. Thus BVA (i) requires that the total volume stay strictly between zero and infinity. Similarly BVA(ii) requires that the total energy stay strictly between zero and infinity. Physical interpretation: these assumptions require the mechanism producing the boundary velocity to be physically realistic. That is, the mechanism causing v, is not allowed to have the infinite power required to push the material to zero or negative volume nor expand it to infinite volume. Nor is the mechanism allowed to have the infinite power required to make the total energy go to zero or negative nor to do an infinite amount of work on the material. Note that BVA(0)
Discrete hydrodynamical model
963
implies the existence of Vol,. It will be seen in the proof of Lemma 2 that the existence of Vol, implies the existence of Erg, through an increasing-entropy argument; similarly the existence of Erg, implies the existence of Vol,. Thus the boundary velocity assumptions reduce to: assume v,(s) is continuous and assume the existence of Erg, on [0, T]. We can also show for certain material laws that assumptions similar to the existence of Vol, imply the existence of Erg,. For example, in an ideal gas equation of state with Navier-Stokes viscosity if the volume of b,_, , pr] is not collapsed by u, then we can show that Erg, exists by using Gronwall’s inequality and a variation of the argument of the proof of Lemma 2. We save the detailed analysis and discussion of these boundary value assumptions, generalizations and relations for a future paper. Lemma
1: For 0
de Ti; (l3 Pj+
I/*) =
c(t,
Pj+
112) g
ft3 Pj+
(27)
l/2).
The proof follows from (4, 7, 14, 17). Lemma 2: Let 0 < T, < T and for t c [0, r,] assume x’(t) = F(t, x(r)) x(O) = %~wl,
CO,&)
x(t) E u. Then there exist positive and finite 6, C, K which do not depend on T, such that x(t)~ S(6, C, K) for all t E[O, T,].
(28)
Proof: For F = V or E let
whereAP,+I,2
=
I++
I -
P,,
First let F = E and observe that ZE(t) is the total energy of the material interval bo, p,] at time t and
Therefore
and thus Zr(t) is the same as the Erg’b,, p,] of (25) and hence Erg’b,, p,] is the total energy of material interval bo, ~~1 at time t. For all t ~10, T], by BVA(ii) 0 < Erg,,, G ZEc,) < Erg, G co. Therefore,
(32)
for t E [0, T, ] and 0 6 j < r - 1 0 < EC!, /++ 112) A/++
I/Z d
Erg,
(33)
and since, for I E [0, T, ] 0 < 46 Pj+ 112) G we
E(t,
Pj+
112)
have 0 < e(t, /++ w) A/++ ~2
G
Erg,
(34)
and 0 < u2(t, Pj+ 112)APj+ 112<
2 Erg,
it follows that there exist finite, positive C, and K (independent
(35) of T,) such that for all
D. L. HICKSand K. L. KUTTLER
964
j=O,...,
r - 1 and t E [0, T, ] 0 < e(t~ Pj+ 112)G cl
and 0 < J”(t, PI+ li2)J G
K.
Next let F = V, observe that Iv(t) is the total volume of [pO,p,] at t and (36) Therefore ,
(37) do) dr s0 and thus r,,(r) is the same as the Vol’[h, ~~1of (23) and hence Vol’bo, pu,]is the total volume of the material interval [pO,p,] at time t. By BVA(i), for all t E [0, T] Ml 1 = MO) +
0 < Vol, < Iv(t) < Vol, < cc.
Therefore,
for t E [0, T, ] and 0 G j < r - 1 0 < v(tv Pj+ 1/2)4,+ I,Z6 VO~M.
Hence there exists a finite, positive Cz (independent
of T,) such that for j E (0, . . . , r - lf and
t E 10, T, 1 O< v(r3Pj+I/2) d
c2.
If C = max(C,, C,) then for 0 Q t G T, max(V(h
14:~)~ . . . , V(h P,- 1,2), dt, PN), . . . , e(T, A- li2 )) G C.
Let K(e)=
Y
c d< , k~
and
H(V)=
, 4Odt
s
s
(38)
and Y(fY Pj+.li2)
=
K(e(f3
P,++I/Z)) +
HtV(r9
Pj+1/2)).
(39)
Y is called the generalized entropy. From Lemma 1, (38) and (39) it follows that the generalized entropy is nondecreasing, i.e. (40) for 0
Pj+1!2)
there exists an Ym such that
- CE< y, - K(C) G H( v(f,
Pj+
1,211
(41)
- CQ< 9, - H(C) G K(e(t,
Pj+
10))
(42)
and
By (13) there exists a 6 (independent
of T,) such that for 0 G j G r - 1
0
Pj+
1~2): 46
Pj+
1,2)).
(43)
This finishes the proof of Lemma 2. Observe that S(6, C, K) is a compact subset of R” and U is an open subset of R3’ containing w, C 0
Discrete hydrodynamical model
965
Lemma 3: Let U c R” open, F(v) -): [0, T] x U +R” be continuous, K, G U be compact, and y > 0 be such that K2 = K, + N(0, y)z U. Let Mr = max((IF(t, x)/l: (t, x)c[O, r] x K,) and let
O
(44)
then there exists a solution to , x(t) = x, +
s0
F;(T, x(r)) dt
with (45)
x(t) E KZ forO
The proof of Lemma 3 is a well-known adaptation of the proof of the Cauchy-Peano 171. Theorem 1: If &, C,,, K. are finite and positive then there exists a solution to
theorem
x’(r) = F(t, x(t)), r E [O,Tl x(O) = x,eS(&,, such that x(t)~S(6,
(46)
C,,, Ko)
(47)
C, K) for some finite, positive 6, C, K.
Proof: By Lemma 3 there exists an 1 > 0 such that if X,E S(6, C, K) then there exists a solution to I x(r) =x,
+
VT, 4~)) dr (48) sU with x(t) E U for 0 < a < t < a + I < T. Therefore for a = 0, there exists a solution to (48) such that x(t) E U for 0 < t < 1. By Lemma 2, x(t) E S(6, C, K). By Lemma 3 again, there exists a solution to (48) with a = I and x, = x(l) for r E[I, 211 with x(r)c U. By Lemma 2 again, x(r)ES(G, C, K). Therefore, there exists a solution to (48) with a = 0 for 0 ,< r ,< 21with x(r) E S(6, C, K). Continuing in this way, Theorem 1 is proved. Observe that only the continuity of F and the a priori estimates were used in Theorem 1 to establish global existence. In Theorem 2 the Lipschitz condition (22) on F is used to globally establish uniqueness and continuous dependence of the solution on the data. Theorem 2: Let x,, and ji, be two physically acceptable vectors and let I x(r) = x0 +
F(z, x(t))
dt
s0 and a(r) = 9, +
’ F(r, 9(s)) dr s0 be two solutions existing by Theorem 1. Then there exists a finite, positive constant L such that
(49)
IIx(r) - W)ll < 1)x0- Ro/exp(W for all r E [0, T].
Proof: By Lemma 2, there exist positive, finite constants 6, C, K such that x(r) and a(r) are both in S(S, C, K) for all r E[O, T]. Therefore, by (22).
/x(t) - W) /I s /lx0 - go 11+ Jo’ IiF(r, ~(4) - WY W)ij
d7
G l\xo-goI) +I;L(6,
C, K)IIW-f(d~~
dt.
(50)
Let L = L(6, C, K) and (49) follows from (50) via Gronwall’s inequality. This proves Theorem 2. M.C.“. IZ,s--D
966
D. L. HICKS and K. L. KU~TLER
Theorem 2 implies that the solution whose existence was proved in Theorem 1 is unique and depends continuously on the data. Thus well posedness is established globally. The inequality in (50) also shows that Model D is not overly sensitive to the choice of the mesh because small changes in pO, . . . , pr will produce small changes in x0 for physcially reasonable data. In summary, we have shown that Model D leads to globally well posed problems when the initial data is physically acceptable and the boundary conditions are determined from given time dependent velocities which are continuous and also required to be physically acceptable. This is proved when the material law is a generalization of the case where the equation of state is of the ideal gas form with a viscosity of the Navier-Stokes form. REFERENCES I. B. Mandelbrot, The Fractal Geometry of Nature. W. H. Freeman, New York (1983). 2. R. P. Feynman, The Character of Physical Law, MIT Press. Cambridge, Mass. (1965). 3. D. L. Hicks and K. L. Kuttler, Continuum and discrete hydrodynamical models and well posed problems. J. Appl. Math. Comput. 25(4), 299-320 (1988). 4. B. L. Keyfitz, Review of shock waves and reaction diffusion equations. Math. Mon. 93, 315-318 (1986). 5. M. Zak, Deterministic representation of chaos with application to turbulence. Mathl Modelling 9, 599612 (1987). 6. K. L. Kuttler and D. L. Hicks, Initial-boundary value problems for Model D (a discrete hydrodynamical model) with stress-boundary conditions are globally well posed. Ma/h1 Comput. h4odelling (in press). 7. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955).
0895-7177/89 $3.00 + 0.00 Copyright 0 1989 Maxwell Pergamon Macmillan plc
INITIAL-BOUNDARY VALUE PROBLEMS FOR A DISCRETE HYDRODYNAMICAL MODEL (MODEL D) WITH VELOCITY BOUNDARY CONDITIONS ARE GLOBALLY WELL POSED D. L. HICKSand K. L. KUTTLER Department
of Mathematical
Sciences, Michigan
Technological
(Received November 1987; accepted Communicated
University,
for publication
Houghton,
MI 49931, U.S.A.
November 1988)
by X. J. R. Avula
Abstract-A discrete model for the one-dimensional motion of a hydrodynamical material is globally well posed when the boundaries have time dependent velocities imposed. The proof is given for the case where the material law is a generalization of the ideal gas form with a viscosity of the Navier-Stokes form. This modelling approach suggests a simple solution to the problem of the deterministic modelling of turbulent motions.
INTRODUCTION The standard, continuum model for hydrodynamical motion we refer to as “Model C”. For many years (going back at least to Perrin in 1906 [l]) some researchers have th;ought that the derivation of Model C from the first principles of classical, Newtonian physics involves some questionable steps [l-3]. One problem is that the molecular nature of matter makes the modelling of a material as a mathematical continuum somewhat mysterious. The classical argument used in deriving Model C involves three stages; (i) assume the material is partitioned into chunks (a chunk of material is a finite set of molecules) “large enough to behave as a continuum in some sense”; (ii) apply first principles to produce equations of motion for the chunks; (iii) take a limit as the chunk size goes to zero to generate the differential equations of Model C. Paradox: how can a finite set of molecules simultaneously be “large enough to behave as a continuum” but have its size go to zero? Model C has been used for many years with seemingly useful results. This may be at least partly an illusion. That is, the discrete analogs of Model C used in numerical methods (finite difference, finite elements, etc.) have apparently been responsible to some extent for this perception of the successfulness of Model C. There are many open questions in the the&y of global well posedness of problems arising from Model C, especially when the initial-boundary value data and material laws are physically realistic. We wonder if there is a connection between these open questions and the questionable derivation of Model C. Global well posedness theory should form the foundation for the study of the continuum and discrete equations representing the models for the motions of materials such as liquids, gases, etc. Unfortunately global well posedness questions for Model C seem to be extremely difficult and good results are few [4]. In a recent paper [3], we presented a discrete model, which we call Model D, that avoids the dubious derivation described above. The Model D approach leads to a system of equations for the evolution of the average energy, momentum, and volume of chunks of the material. Model D is derived by applying the first principles of classical Newtonian physics to chunks or finite cells of the material. If we were to give the method of deriving Model D a name, we might call it a “finite cell method”. Although Model D is not derived by applying a finite difference or finite element method to Model C, it does resemble Model C with derivatives discretized, of course. Global well posedness results are important in various numerical analysis problems that arise in computational continuum dynamics. For example, the analysis of convergence and error estimates is much easier if global well posedness has been established. Well posedness is important from a deterministic modelling perspective because in this discipline a single solution should exist and depend continuously on the initial and boundary data. 959
960
D. L. HICKSand K. L. KUTTLER
In a recent article in the journal Zak [5] presented a nice discussion of the problem of producing deterministic mathematical models for turbulent motions. Quoting from Zak [5]: “As shown by physical and numerical experiments, a complete description of chaotic motions is impractical since such motions are sensitive to small errors in initial conditions, and therefore, are non-reproducible no matter how precisely the experiment is repeated. However, the most striking characteristic of chaotic motions is the stability of certain averaged quantities, such as the mean velocity field of a turbulent motion. Indeed, despite the fact that the velocity at each particular point of a turbulent flow is unstable and unpredictable, the mean velocity profiles are usually stable and reproducible. . . . In view of these ideas it is reasonable to express the original mathematical models in terms of stable and reproducible quantities of chaotic motions”. We suggest that the Model D approach provides a simpler solution to the problem of the deterministic mathematical modelling of turbulent motions. We prove that Model D provides a stable, deterministic model because that follows, of course, from the global well posedness. Although this paper presents results only for the one-dimensional version of Model D, these results can be extended to higher dimensions as will be shown in future papers. What has been established for Model D: in [3] we presented the derivation of the model and showed that a problem involving the response of a material interval with both ends fixed was globally well posed. In [6] we showed global well posedness for the stress boundary case. In this paper global well posedness is proved for the case where velocity boundary conditions are prescribed. This is done for a material law that generalizes the case when the equation of state is of the ideal gas form with viscosity of the Navier-Stokes form. The initial data is assumed to be physically acceptable. The details are developed here only for the case where one end is fixed, but the case with nonzero time-dependent velocities prescribed on both ends can be analyzed in a similar manner. To begin, Model D is presented and then the global existence, uniqueness, and continuity of the solution operator is established. MODEL
D WITH
VELOCITY
BOUNDARY
CONDITIONS
Let the material interval be [pO,~~1 and to it assign a partition 0 = PO< p, <. . . < p,. Let the cross sectional area of the one-dimensional interval [cl,, pi+ ,] is
(1)
material be A. Then the mass of the material
Massbj7~,+Il=A(~,+I -II,).
(2)
It is convenient to take A = 1. Let V, u, E denote specific volume, specific momentum, specific total energy. Vol’[p,, cl/+,] is the volume of [p,,, pj+ ,] at time r. The average specific volume of [p,,, .P~+,] at time t is given by v(t, Pj+~~2)=Vol’[(Pj~ /++~llMN~~~Pj+ll
(3)
and E(t, pj+ ,!2), u(t, pj+ ,,*) are similarly defined. Model D, derived in [3], is given by g
(t, P,+ I?) =
u(r9P~+ I) - u(t3PjLi) Pj+l
g
(t, P,+ Iid =
-PI
E(t, P,+ I>- Z(t9 P,) Pj+l-Pj
$ ft3
Pj+
12)=
w(t3Pi+ I) - w(tYPj> Pj+I-II,
(4)
Discrete hydrodynamical model
961
for o
‘ttP Pji> = t”ttv Pj+ 19)+ u(t9Pj- I/*))/2 ‘tt,
(8)
Pj) = tctt2 Pj+ IL?)+ c(t, Pj- l/2)1/2
(9)
w(t, PjFci) = t”ttY Pj+ 1;2)C(t2Pj- l/7_) + utt, PI- IQ)C(t, /$‘I+IQ))/2 for
I sgj G r - 1. The stress Z(t, p,+ ,,?) is given by
with the specific internal energy e given by e(tY l(j+l/*)
=
tE
-
u2/2Nt,
Pj+
(11)
I/Z).
In equation (10) P is the pressure and 1 >, 0 is the coefficient of viscosity. P is given by a generalization of the ideal gas law: P(e, V) = k(e)h( V)
where k and h are positive for e and V positive and
(12)
s
’ de
S’
h(V)dV=co=
(13)
Ok(e)'
0
Recall that the ideal gas law is P(e, V) = Te/V where r > 0 is a constant. For the ideal gas law k(e) = Te and h(V) = l/V’; note that equation (13) is satisfied by the ideal gas law. In the Navier-Stokes viscosity A = A,p where I, > 0 is a constant. Thus the material law considered here is a generalization of the case where the equation of state is of the ideal gas form and the viscosity is of the Navier-Stokes form. Lipschitz conditions are assumed on h, k and Iz in all arguments for 0 < 6 < min(e, V) G max(e, V) 6 C < cc and 1~1< K c 00; the Lipschitz conditions may depend on 6, C and K. Note that no convexity assumptions are made on h, k or 1. The unknown functions in equation (4) are V, U, E at (t, pj+ ,,2) for 0
PO)
=
Z(t,
(14)
A,2)
a(t, PO)= 0
(15)
w(r, &I) = 0
(16)
at the right boundary Z(t, /A) = Z(t, P,-
(17)
l/2>
u(t, P,) = u,(t)
(18)
W(GP,) = a(t, PL,)C(tvcl,).
(19)
These conditions represent the left end of the material as fixed while the right end experiences a time dependent velocity u,; assume u, is continuous. Letting (x0, . . * 3X,-I, x,, . . . 3&-1,X2,,
* * . , X3-l) =
(Vf., Et.9
h/2)9 Pl,2),
. * * 7 V(. . . . 3 Eta
9 /A3 A-
1,2), UC.
7 Pl,Z),
' . * 9 4'3
ru,- I/Z)>
I/Z))
the system of differential equations (4) for Model D with conditions (7)-(17) may be written as x’(t) = F(t, x(t)).
(20)
D. L. HICKS and K. L. KUTTLER
962
LetS(6,C,K)={xER3’:O<6bmin(x, ,..., x,_,,yZ ,,..., y,, _ , ) d C and max(lx,I, . , jxz, _ , /) < Kj where y*l+i= Note
y,,_,)
.
r.~,_l,+YZ,, . . . ,
X2+, - (x”+,)*P
that Y2r+t = e(t,
i.e. y is the specific internal Let
!A+ 1;21r
energy.
U = U(S(6,
C, K):O < min(6,
C. K) Q max(b, C, K) < co}.
U is called the set of &&ally acceptable vectors. Note that the V, U, E which are constructed from components of the vectors in U are such that V, e are positive and V, (~1, e are bounded. If the initial data are in U then they are said to be physically acceptable. Note that the only requirement on the initial data is that it be physically acceptable. Let x(0) = x0
(21)
and 5, = (V:,,, . . . , V:_ ,,I, u::,, . . , u;_ ,;2, E?;2, .
. , E;_ ,iz).
Suppose
with 6,, C,, & strictly positive and finite and 6, < C,. Thus the initial V and e are between So and C, and the initial u is between -K, and + K,. The assumptions on h, k, ,l imply that for x, y E S(6, C, K), there exists a Lipschitz constant L(6, C, K) such that IjFk XI - F(r, Y>II6 U&
C, K)\jx - YI(
and the continuity of C, implies that F is continuous in t. The boundary velocity assumptions are as follows. BVA(0): cr(.) is continuous on [0, T]. BVA(i): let VO~~[~~,p,] be the total volume of [po, p,] at time t = 0. Let Vol’[po, p,] be defined
(24
by
I VOl’[Po, Prl = vo~“[Po, i&l+ Then
s0
Gz> dz
(23)
< cc,.
(24)
for all t in [0, T], 0 < Vol, < vol’[~o) pr] < Vol,
BVA(ii):
let ErgO[p,, p,] be the total energy of [pO, p,] at time t = 0. Let Erg’[p,, FL,]be defined
by
I Wbo7 P,I = Ergok0TAI+ Then
s0
OAT P (z>clr 1dr.
(25)
for all t in [0, T] 0 < Erg, < Erg’[po, ~~1 Q Erg,
< 00.
(26)
In the proof of Lemma 2 it is shown that Vol’[p,, p,] is the total volume and Erg’[po, p,] is the total energy at time t of [po, IL,]. Thus BVA (i) requires that the total volume stay strictly between zero and infinity. Similarly BVA(ii) requires that the total energy stay strictly between zero and infinity. Physical interpretation: these assumptions require the mechanism producing the boundary velocity to be physically realistic. That is, the mechanism causing v, is not allowed to have the infinite power required to push the material to zero or negative volume nor expand it to infinite volume. Nor is the mechanism allowed to have the infinite power required to make the total energy go to zero or negative nor to do an infinite amount of work on the material. Note that BVA(0)
Discrete hydrodynamical model
963
implies the existence of Vol,. It will be seen in the proof of Lemma 2 that the existence of Vol, implies the existence of Erg, through an increasing-entropy argument; similarly the existence of Erg, implies the existence of Vol,. Thus the boundary velocity assumptions reduce to: assume v,(s) is continuous and assume the existence of Erg, on [0, T]. We can also show for certain material laws that assumptions similar to the existence of Vol, imply the existence of Erg,. For example, in an ideal gas equation of state with Navier-Stokes viscosity if the volume of b,_, , pr] is not collapsed by u, then we can show that Erg, exists by using Gronwall’s inequality and a variation of the argument of the proof of Lemma 2. We save the detailed analysis and discussion of these boundary value assumptions, generalizations and relations for a future paper. Lemma
1: For 0
de Ti; (l3 Pj+
I/*) =
c(t,
Pj+
112) g
ft3 Pj+
(27)
l/2).
The proof follows from (4, 7, 14, 17). Lemma 2: Let 0 < T, < T and for t c [0, r,] assume x’(t) = F(t, x(r)) x(O) = %~wl,
CO,&)
x(t) E u. Then there exist positive and finite 6, C, K which do not depend on T, such that x(t)~ S(6, C, K) for all t E[O, T,].
(28)
Proof: For F = V or E let
whereAP,+I,2
=
I++
I -
P,,
First let F = E and observe that ZE(t) is the total energy of the material interval bo, p,] at time t and
Therefore
and thus Zr(t) is the same as the Erg’b,, p,] of (25) and hence Erg’b,, p,] is the total energy of material interval bo, ~~1 at time t. For all t ~10, T], by BVA(ii) 0 < Erg,,, G ZEc,) < Erg, G co. Therefore,
(32)
for t E [0, T, ] and 0 6 j < r - 1 0 < EC!, /++ 112) A/++
I/Z d
Erg,
(33)
and since, for I E [0, T, ] 0 < 46 Pj+ 112) G we
E(t,
Pj+
112)
have 0 < e(t, /++ w) A/++ ~2
G
Erg,
(34)
and 0 < u2(t, Pj+ 112)APj+ 112<
2 Erg,
it follows that there exist finite, positive C, and K (independent
(35) of T,) such that for all
D. L. HICKSand K. L. KUTTLER
964
j=O,...,
r - 1 and t E [0, T, ] 0 < e(t~ Pj+ 112)G cl
and 0 < J”(t, PI+ li2)J G
K.
Next let F = V, observe that Iv(t) is the total volume of [pO,p,] at t and (36) Therefore ,
(37) do) dr s0 and thus r,,(r) is the same as the Vol’[h, ~~1of (23) and hence Vol’bo, pu,]is the total volume of the material interval [pO,p,] at time t. By BVA(i), for all t E [0, T] Ml 1 = MO) +
0 < Vol, < Iv(t) < Vol, < cc.
Therefore,
for t E [0, T, ] and 0 G j < r - 1 0 < v(tv Pj+ 1/2)4,+ I,Z6 VO~M.
Hence there exists a finite, positive Cz (independent
of T,) such that for j E (0, . . . , r - lf and
t E 10, T, 1 O< v(r3Pj+I/2) d
c2.
If C = max(C,, C,) then for 0 Q t G T, max(V(h
14:~)~ . . . , V(h P,- 1,2), dt, PN), . . . , e(T, A- li2 )) G C.
Let K(e)=
Y
c d< , k~
and
H(V)=
, 4Odt
s
s
(38)
and Y(fY Pj+.li2)
=
K(e(f3
P,++I/Z)) +
HtV(r9
Pj+1/2)).
(39)
Y is called the generalized entropy. From Lemma 1, (38) and (39) it follows that the generalized entropy is nondecreasing, i.e. (40) for 0
Pj+1!2)
there exists an Ym such that
- CE< y, - K(C) G H( v(f,
Pj+
1,211
(41)
- CQ< 9, - H(C) G K(e(t,
Pj+
10))
(42)
and
By (13) there exists a 6 (independent
of T,) such that for 0 G j G r - 1
0
Pj+
1~2): 46
Pj+
1,2)).
(43)
This finishes the proof of Lemma 2. Observe that S(6, C, K) is a compact subset of R” and U is an open subset of R3’ containing w, C 0
Discrete hydrodynamical model
965
Lemma 3: Let U c R” open, F(v) -): [0, T] x U +R” be continuous, K, G U be compact, and y > 0 be such that K2 = K, + N(0, y)z U. Let Mr = max((IF(t, x)/l: (t, x)c[O, r] x K,) and let
O
(44)
then there exists a solution to , x(t) = x, +
s0
F;(T, x(r)) dt
with (45)
x(t) E KZ forO
The proof of Lemma 3 is a well-known adaptation of the proof of the Cauchy-Peano 171. Theorem 1: If &, C,,, K. are finite and positive then there exists a solution to
theorem
x’(r) = F(t, x(t)), r E [O,Tl x(O) = x,eS(&,, such that x(t)~S(6,
(46)
C,,, Ko)
(47)
C, K) for some finite, positive 6, C, K.
Proof: By Lemma 3 there exists an 1 > 0 such that if X,E S(6, C, K) then there exists a solution to I x(r) =x,
+
VT, 4~)) dr (48) sU with x(t) E U for 0 < a < t < a + I < T. Therefore for a = 0, there exists a solution to (48) such that x(t) E U for 0 < t < 1. By Lemma 2, x(t) E S(6, C, K). By Lemma 3 again, there exists a solution to (48) with a = I and x, = x(l) for r E[I, 211 with x(r)c U. By Lemma 2 again, x(r)ES(G, C, K). Therefore, there exists a solution to (48) with a = 0 for 0 ,< r ,< 21with x(r) E S(6, C, K). Continuing in this way, Theorem 1 is proved. Observe that only the continuity of F and the a priori estimates were used in Theorem 1 to establish global existence. In Theorem 2 the Lipschitz condition (22) on F is used to globally establish uniqueness and continuous dependence of the solution on the data. Theorem 2: Let x,, and ji, be two physically acceptable vectors and let I x(r) = x0 +
F(z, x(t))
dt
s0 and a(r) = 9, +
’ F(r, 9(s)) dr s0 be two solutions existing by Theorem 1. Then there exists a finite, positive constant L such that
(49)
IIx(r) - W)ll < 1)x0- Ro/exp(W for all r E [0, T].
Proof: By Lemma 2, there exist positive, finite constants 6, C, K such that x(r) and a(r) are both in S(S, C, K) for all r E[O, T]. Therefore, by (22).
/x(t) - W) /I s /lx0 - go 11+ Jo’ IiF(r, ~(4) - WY W)ij
d7
G l\xo-goI) +I;L(6,
C, K)IIW-f(d~~
dt.
(50)
Let L = L(6, C, K) and (49) follows from (50) via Gronwall’s inequality. This proves Theorem 2. M.C.“. IZ,s--D
966
D. L. HICKS and K. L. KU~TLER
Theorem 2 implies that the solution whose existence was proved in Theorem 1 is unique and depends continuously on the data. Thus well posedness is established globally. The inequality in (50) also shows that Model D is not overly sensitive to the choice of the mesh because small changes in pO, . . . , pr will produce small changes in x0 for physcially reasonable data. In summary, we have shown that Model D leads to globally well posed problems when the initial data is physically acceptable and the boundary conditions are determined from given time dependent velocities which are continuous and also required to be physically acceptable. This is proved when the material law is a generalization of the case where the equation of state is of the ideal gas form with a viscosity of the Navier-Stokes form. REFERENCES I. B. Mandelbrot, The Fractal Geometry of Nature. W. H. Freeman, New York (1983). 2. R. P. Feynman, The Character of Physical Law, MIT Press. Cambridge, Mass. (1965). 3. D. L. Hicks and K. L. Kuttler, Continuum and discrete hydrodynamical models and well posed problems. J. Appl. Math. Comput. 25(4), 299-320 (1988). 4. B. L. Keyfitz, Review of shock waves and reaction diffusion equations. Math. Mon. 93, 315-318 (1986). 5. M. Zak, Deterministic representation of chaos with application to turbulence. Mathl Modelling 9, 599612 (1987). 6. K. L. Kuttler and D. L. Hicks, Initial-boundary value problems for Model D (a discrete hydrodynamical model) with stress-boundary conditions are globally well posed. Ma/h1 Comput. h4odelling (in press). 7. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill, New York (1955).