DIFFERENCE SCHEMES FOR WEAK SOLUTIONS OF MIXED PROBLEM FOR A CLASS OF HYPERBOLIC DIFFERENTIAL EQUATIONS, I HOANG DINH DUNG, TRAN XUAN BO Institute of Mathematics, Vietnam Abstract.. In
Trang 1DIFFERENCE SCHEMES FOR WEAK SOLUTIONS
OF MIXED PROBLEM FOR A CLASS
OF HYPERBOLIC DIFFERENTIAL EQUATIONS, I
HOANG DINH DUNG, TRAN XUAN BO
Institute of Mathematics, Vietnam
Abstract It is known that many applied problems are reduced to mixed problems of hyperbolic differential equations with nonregular data The approximate methods for these problems are studied
by some authors For example, in [1-3] are considered the cases of data belonging to the Sobolev spaces W7" (Q) In [4] the convergence rate of approximate solution for the mixed problem is obtained
by the method based on norm estimates in the Sobolevic spaces H™°(Q) In this paper we propose a method to extend the ideas introduced in [5, 7] for investigating the approximate solutions of mixed problem for the hyperbolic differential equations with variable coefficients in the space H™(Q) (see sec 2) In section 3 it is first time this approximate problem is considered in the space of generalized
functions D'(Q) > we™ (Q)
Tóm tắt Nhiều bài toán ứng dung được đưa về dạng bài toán của phương trình hyperbolic véi dir liệu không trơn Trong [1+4] đã xét các bài toán với dữ liệu thuộc các không gian Sobolev W/7"(©) Còn trong bài báo này, chúng tôi tiến hành nghiên cứu nghiệm xấp xỉ các bài toán có dữ liệu không
trơn độ cao, cụ thể là thuộc các không gian Schwartz ?'(Ó) 5 W7"(©)
1 INTRODUTION
Consider the initial and boundary value problem for the following class of hyperbolic differential equations:
2
where the coefficients k;(a) € C1(G), ki(x) => C > 0, i = 1,2, C is a constant, G is a bounded
region in R?,Q=Gx (0,7) = {(#,t): cE G,0<t<T< ow}, =0G x (0,7)
Suppose that the data f(x, £), ø() are not continuously differentiable in the classical sense
In these cases the generalized solutions (GS) are considered Below, at first, we consider GS
u of the problem (1)-(3) in the Sobolev spaces H™(Q) with the corresponding test functions
v defined in the spaces Ởm(Q), m being the nonnegative integers
* This publication is completed with financial support from the Council for Natural Sciences of Vietnam and by the Program “Applied Mathematics” NCST of Vietnam.
Trang 2By [8] the GS u(x,t) of the problem (1)-(3) satisfies the condition (3) and the following integral equality (wu is extended by zero onto Q~ = R(t) x R?(x) \ Q):
= | U69) + HG, Cado(Gs,Ca,o)]ae— f o(6s,6a)
Q
G
Øu(€)
C3 ¢3=0
where u(€) = u(G, Ca, G3), p(x) = p(x, x9), u(x, t) = 0(1, £2, ts), v(¢) = v(¢1, Ca, G3)
2 DIFFERENCE SCHEME FOR GS
2.1 Construction of difference scheme
For simplicity of presentation, let Q be the unit cube:
Q = {(x, t) = (#1, vo, t3) : 0 < a1, v0, t3 < 1} Let us introduce in © a grid w:
w= { (1, #2, ts) 1, = iy, = Jali, tg = jghs; 7, = 0,1, , Nis,
hị Các fS— 1,3 ý — 0/1, M, hạ = Oh,
where N; and M are positive integers For the steplengths h;, 7 = 1,2,3, assume that C, <
= < Co, C3 < Đế < Cy uniformly as hị, hạ, hạ — 0, here Ở,, m — 1,4, being positive constants
Denote the set of interior and boundary gridpoints of Q by w and y respectively, y =@\w
To obtain a net problem we introduce an auxiliary cubic grid covered the cube 2 and containing three families of planes which are parallel to the coordinate planes x,Ox2, x20ts, tg3Ox, with steplength distances hy, ho, hg respectively In 0 denote by œ* this grid consisting
of the parallelepipeds with centres at the gridpoint (,t) of the grid w The cell of w* at the gridpoint (x,t) is denoted by e:
c— {¢ = (C1, C2, €3) : |& — 4] < 0.5 hị, t= 1,2, re — ts| < 0.5 hg}
Now, as in [5] (see Sec 1, Chap 3) one may take the test function v in (4) by the form
(w t) — { (hy ho hạ) 1 for (z, t) cc,
Then, by (4), the GS u of the problem (1)—(3) satisfies the following integral equality:
Let us set
uy, sey Gos 5 Yn) da 5
Trang 3yitke) = yithe) (y) = UY1, 5 Yat kha, Yn);
tụ, =,() = TT TT, ug, Sg, y) =“
where l <œ<w, k=0,5,1; , cC", here n=3
Then, from (6) we have the following net problem for the GS u(z, t) of the problem (1)-(3):
P°u = SISa5s— + 193 52 — 5 SsS%_¿ | k¿=— 33 ( mì =R ƒ#+ Su, 6U, (z,†) (x,t Cœ , (7)
u=0, (x, t) ey,
where R= S,S9S3, S= S162 :
To obtain the difference schemes of the operator P®u (7), one may approximate the mean integral operator S;, 7 = 1, 2,3, by the quadrature formula of average rectangles and the partial derivatives by difference quotients as in [7] For instance, one has
Ou (0.51) S352 | ky —
t3+0.5h3 £2+0.5h2
= ky (xy — 0.001, 2) (a, — 0.001, 22; 23)dzodz3
t3—0.5h3 v2—0.5h2
—0.5
~ kị SD yn
1"
Then, one has the following difference scheme of the problem (1)-(3)
2
y(w,t) =0, (@,t) € 9, where g = Rf + Sy
The difference schemes of the form (8) are investigated by many authors (see, e.g., [6]) The scheme (8) may be written in the form:
2
Phy = $1 5>Sayp.,— > (MP ye.) = glersva,ta), (wt) ew, ©)
y(w,t) = 0, (a, t) 4,
2.2 Estimation of the convergence rate
Consider now the convergence of the approximate solution y to the GS wu of the form
(4), (5) of the problem (1)-(3) For this purpose we estimate the method error z = y — u of the scheme (9) From (9) one has
2
` Kh ODay
(0),
Hence, by (7) and (9),
Lz=p—lu=—WV(z,t), (x,t) Ew,
2(x,t) =0, (w,t) © 4,
Trang 4
where
W= > (x! wz) - > S55 ¡ ( a) + $1 S553 (Sa thes)
Thus,
— Do k 085 (Aan), = Dolla, +A 2 ) = t + ^ ’ st 0) ee cứ, (10
2(x,t) =0, (x,t) € 4,
where
h = kì um, = $353_; (52) 3 Ao = SaSa5%3 ace — Yes : (11)
Now, to obtain the error estimation, consider the space H of grid functions u on & and let Ho be its subset of the functions satisfying the condition u(z, t) = 0 as (x,t) € 4
Let a(z,t), b(@,t) € Ho or H Introduce the following scalar products and corresponding grid norms:
(a,b) = S" a(w,t)b(a, t)hihohs ,
(œ,t)€œ MeN, No-l
(a, 7? = » » » alii hy, the, jahs)b(is hi, igha, jshs)hihohs ,
j=0 i1=1 ig=1
M Ni-1 Np (a, 2 = » » » alii hy, the, jahs)b(is hi, igha, jshs)hihohs ,
j=0 i= ig=l
llall? = llello = (4), llallio = @.al??, #= 1,2
Let us scalar multiply both sides of (10) with z(z, t):
2
-Ÿ (0 292),2) ~Ê 6,213
¿=1
Then, by the same way as In Sec 2.2 [7| one has
lz|Ìi < Œ (>: llinlo + Da) (12)
=1
where the constant C is independent of h and 2 (|h|? = hi + h3 + h3),
2 20"
2
Izll: - = llzllễ„ + IIVzll IIVzllô.„ = Ð › llzz.l
=1
To estimate the terms in the right-hand side of (12), we first consider the functional 7 (z, t)
defined by (11):
1 t3+0.5h3
m (x,t) = ky (x, — 0.5h1, ma}ug, — x | x
3 Jta—0.5ha
œa-L0.Bha
_— ky (xy — 0.001, 65) — (#1 — 0.001, Ca, €3)d€o d¢3
hạ 43 —0.5h2 aC,
Trang 5We see that the expression of 7;(x,¢) is anologous to the one of m in Sec 2 [4], then by the same way as we did for the estimation (26) in that section, one has
llislio < C|h|7—*lells, 2— 1,2, m= 2,3, (13)
where
1/2
Iali».e = lIellz=eœ = | 3 [ D*u(w, t) 2 dedt
|a|<m
For the term \o, by (11) one has
Ou
Ao = S$ S253 (Se — ty ) + $1 S293 (7,1, — „¡„) = Ào + ÀF (14)
By the Cauchy—Buniakovskij inequality one has
IAjI < (hịhạha)—'/°{ / l2 _ 1, (2, 1)| ‘acy
9Œ
One has
Hãa = 7,608) d¢ =
2
t3 ++ha
OPu(C) A? ula, Lo, a)
Then,
JAG? < C(hihahs) "hl? (ul3.e + lel3 eg)
where
€3 = es(œ, £) = {¢ = (C1, C2, €3) : |& — ¿| < 0.0h¿, + — 1,2; ts — hg < G < tạ}
1/2
|a|=m e
Thus,
From (14) and (15) it follows
I|Aol] < C1Al [lulls,0 (16)
Combining (12), (13) and (16) yelds
lz|Ì = lly - wlio S ClAl llulls,o (17)
Further, for the problem (1)-(3) one has the following a priori estimate (see [10,Sec 2, Chap 5]):
I«||s.ø < C(lells.o + ll¿lls + l/|s,o)):
Trang 6where the constant C is independent of y, 7 and f
Finally, from the last inequality and (17) one has the following
Theorem 1 Let the given functions f © H?(Q), c H2D(G), » © H°(G) and ki(x) < W2 (G)n C(G), i=1,2 Then the solution y of the difference scheme (8) converges to the G'S (6) u(x,t) (u € H3(Q)) of the problem (1)-(3) in the grid norm ||-|l1, with the rate O(\h|), such that one
has the following error estimation
llu — +|Ìi,„ < ClAl lulls.o,
where the constant C is independent of h and u(œ, 1)
3 DIFFERENCE SCHEMES FOR WEAK SOLUTION
Now consider the GS wu(z, ¢) of the problem (1)-(3) in the form (4), where the test functions
has the form
where | is a positive integer
Then, by (4) the GS u(z,t) of the problem (1)-(3) satisfies the following equality:
2
(hị haha) Noe 2C ` = |b (G1, ¢2) 5 | a(g)dg =
(hahaha) " [ {(Oa(Q) + 66 G)a(G.6,0)= e(G:6)2 | lúc, 0)
where a(¢) = hịhaha 0(€)
Thus, one has the following net problem for the GS u(z, ¢):
Pu = 8525 dưng — S55 (số) |
158574 (C1, Ga) => OG; OC; =Rf+Sb-Te =4 (w,t) Cw,
ula, )=0, (x,t) ©, where Rf = $1 $23 a(0)f(¢), Sb = $182 lal, 6, OW(G, ©)],
oa)
Te = S182 loa, 62) oe
3
3.1 Difference schemes
From (20), arguing as in the proof of the form (8), Sec 2.1, we obtain the following difference approximation of the problem (1)—(3):
2
2
Pry = 51 S253 a(¢ W ats ¬¬" d;Jz,) c S$ S253 So kilw)oz, te,
y(x,t) =0, (x,t) €4,
Trang 7where a¿ = a( 95: JR 0:5) tạ,
Furthermore, as in [4] (see the forms (9) and (12) in Sec 2.1) one has an other difference scheme of the problem (1)—(3):
2
° PED = 519253 F741, — D, (bidls.)n, = 4 (x,t) Ew,
9(,0) =0, (@,t) € 9,
where 6;(x) = kệ 952 (zx),
Gl 1) — 81555: » 9383-4 (asso) + 51 S253 Ym €6) ae: (23)
3.2 Estimation of the convergence rate
a) Consider first the scheme (22) We see that the approximation (22) has the form (9), then
by (17),
lứ — +|li,„ < ClAl lulls.o, (24)
where g being the solution of the scheme (22), u being the GS of the problem (1)-(3)
Now consider the following mixed scheme:
1
My =} (PE PPE y =a, (0) ew
y(w,t) =0, (w,t) Ey,
1
where y = = (y+), y and g being defined by (21) and (22) respectively
Then, by (21) and (22), NW]
2
Qy = 5915$5[1 +0) tse, — 5 2~ lí + J0,
2
(œ,£) =0, (œ,t) € +
Note that it can be verified that
lim / _0(©)0(Qd€ — g(e,1), hị,ha,ha—>0
where the function g(¢) is summable in e and continuous at the gridpoint (z, £)
Hence, if hi, ho and hg are sufficiently small, one a write (25) in the form:
Đi, — TM) Ta °°) pal °°) J+ 5 Doheny = q, (x,t) Ew, (26)
i=l
yle,t) =0, (w,t) € 7
By [6] it should be noted that there exists uniquely a solution of the difference problem
(26) and, then, of the scheme (21)
Trang 8Consider now (25), one has
2
y= ` [(ai + bi) ye; Jy, = 515253[1 + a()]0y,
i=l
2
+ $1 S953 ` k;œ, Um, — 2# =ƒÿ, (2, t) Ew (27)
=1
Let 2 =y—u, where u being the GS of the problem (1)-(3) in the form (4), (18) By (27), (23) and (20),
Qoz = ? — Qou = —~(x, t), (x, t) cứ,
28
elu,t) =0, (,9 € +, (8)
where
¬¬ (a, + 04) #Zm, |„ = Soin + tile, + Ao + Ar + 8,
“mẽ mẽ“ ni
3— RŠk 200 Ou 0G AG “T5
From (28) one has the following inequality analogous to (12):
2
IIzIhw se (>: [limlso + l2|sol + [Aol + [Aa + ia)
=1
The terms in the right hand side of the last inequality are estimated by exactly the same manners as they were done above in Sec 2 and as in [7] (see Sec 2) Then one has
Finally, by (24) and (30) we get the estimation of method error for the difference scheme (21):
ll — +||, < ClAl llella.o- (31)
Remark By a manner analogous to the proof of the inequality (31), one may verify that this
estimation is also valid if, in the form (4) of the GS u(x,t) (4), v(z,#) is any test function in
the Schwartz basic space D(M’), 0’ € Q
b) The estimation (31) is obtained with the assumption f € Lo(Q) and y € Le(G), we now
show that the result may be generalized to the equations with right hand side f € D’/(Q) and the initial conditions y, ~ € D'(G), here D’(G) being the space of Schwartz distributions on G
[9].
Trang 9Indeed, by our assumption and by the theorem on local structure of distributions and its
corollary, there exist the functions g € H?(e), s(x) € H?(eo), r(x) € H?(eo) and the nonnegative
integers ky, ke, kz3 such that
f=DEDPPg(a,t), b= D&s(x), oe € DE r(x), (32)
a2*
where DE = 2ahoaE ' the open set e Cc 9 C R(x) x R(t), e = {(x,Ð)
TONS
(x,t) Ee, t =O}
Let v € D(e) By [8], the GS u of the problem (1)—(3) satisfies the following equality:
(ZED (noe) ow) = (Feo x 019-4010) «010.0 (33)
where ø, ƒ, Ở and ¢ are the extended functions of u, f, 7 and y by zero onto QT = R(t) x R?(z)\Q and G~ = R?(x) \ G respectively, (u,v) denotes the value of a functional wu on the basic function v
By (32) one may write (33) as
j\- ar
=
~ |h (¢1, ¢2) a Fe] } (0% = + âu ~fc (34)
where
Rg= / g(C)or (Ode, ; Sb = / s(Q,@}02(Q, @)d€, 7= / r(Ci, @})0z(Q, @)d&,
=
vy (x,t) = (-1)"D™ DPv, vo(x) = DỲ?2o(x,0), v3(x) = Dk [Dre(,Ð], Tạ:
We see that ơ(œ,#), 0a(xz) and v3(x) are also the test funelions: ơi € Đ(€); 0a,a € (so)
Thus, the equation (34) has the form (4), (20) Hence, one may repeat the procedure used above for the difference schemes (21), (22) and obtains the following result analogous to (31): Theorem 2 Let in the problem (1)-(3) the data f © D'(Q), ¢,v © D'(G) and ki(x) c W2 (G),
i=1,2 Then, there exists uniquely a solution of the difference scheme (21) and this solution
y converges to the GS (33), (34) u(x,t) of the problem (1)-(3) with the rate O(|h|) such that
one has the error estimation
| - + < Cl®I llll.o:›
where OQ! €Q
Remark
1° For the sake of simplicity, the homogeneous condition (3) was considered In the case
of nonhomogeneous condition, the theorems 1, 2 may be proved quite analogously
2° In the part II of this publication we will consider the difference schemes of the problem
(1)-(3) in a region of arbitrary form
REFERENCES [1] A Quarteroni, A Valli, Numerical Approximation of Partial Differential Equations, Springer, New York, 1997
Trang 10[2] R Falk, G Richter, Explicit finite element methods for symmetric hyperbolic equations, SIAM J Numer Anal 36 (3) (1999) 935-952
[3] J Trangenstein, Numerical Solution of Partial Differential Equations, Duke Ed., Durham,
2000
[4] T Medjo, Iterative methods for a class of control problems in Fluid Mechanics, STAM J
Numer Anal 39 (5) (2001) 1625— 1647
[5] A.A Samarskij, R.D Lazarov, V.L Makarov, Difference Schemes for Generalized Solu- tions of Differential Equations (Russian), Vus Univ., Moscow, 1987
[6] A.A Samarskij, Theory of Difference Schemes (Russian), Science, Moscow, 1983
[7] Hoang Dinh Dung, Difference schemes for generalized solutions of some elliptic differential
equations, I, Journal of Comp Science and Cybern 15 (1) (1999) 49-61
[8] V.S Vladimirov, Generalized Functions in Mathematical Physics, Mir, Moscow, 1979 [9] L Schwartz, Théorie des Distributions, Hermann, Paris, 1978
[10] V.P Mikhailov, Equations aux Dérivées Partielles, Mir, Moscou, 1980
Received on April 1, 2002 Revised on October 24, 2002