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Explicit schemes for parabolic and hyperbolic equations
Applied Mathematics and Computation 250 (2015) 424–431
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Explicit schemes for parabolic and hyperbolic equations q P.N. Vabishchevich ⇑ Nuclear Safety Institute, Russian Academy of Sciences, 52, B. Tulskaya, 115191 Moscow, Russia North-Eastern Federal University, 58, Belinskogo, 677000 Yakutsk, Russia
a r t i c l e
i n f o
Keywords: Parabolic equation Hyperbolic equation Finite difference schemes Explicit schemes Alternating triangle method
a b s t r a c t Standard explicit schemes for parabolic equations are not very convenient for computing practice due to the fact that they have strong restrictions on a time step. More promising explicit schemes are associated with explicit–implicit splitting of the problem operator (Saul’yev asymmetric schemes, explicit alternating direction (ADE) schemes, group explicit method). These schemes belong to the class of unconditionally stable schemes, but they demonstrate bad approximation properties. These explicit schemes are treated as schemes of the alternating triangle method and can be considered as factorized schemes where the problem operator is splitted into the sum of two operators that are adjoint to each other. Here we propose a multilevel modification of the alternating triangle method, which demonstrates better properties in terms of accuracy. We also consider explicit schemes of the alternating triangle method for the numerical solution of boundary value problems for hyperbolic equations of second order. The study is based on the general theory of stability (well-posedness) for operator-difference schemes. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In the numerical solution of boundary value problems for evolutionary equations, emphasis is on the approximation in time [1–3]. For parabolic equations of second order, unconditionally stable schemes are based on implicit approximations. In this case, we must solve the corresponding boundary value problem for an elliptic equation at every new time level. To reduce computational costs, explicit schemes or different variants of operator-splitting schemes are employed [4,5]. Explicit schemes have evident advantages over implicit schemes in terms of computational implementation. This advantage is especially pronounced in the construction of computational algorithms oriented to parallel computing systems. At the same time explicit schemes have the well-known disadvantage that is associated with strong restrictions on an admis2 sible time step. For parabolic equations, the stability restriction has the form s < s0 ¼ Oðh Þ, where s is the time step and h is the step of the spatial grid [6,7]. Some promises are connected with explicit schemes, where calculations are organized in the form of traveling computations. In fact, such schemes are based on the decomposition of the problem operator into two operators, where only one of them is referred to a new time level. That is why such schemes with inhomogeneous approximation in time are called explicit–implicit schemes. These schemes are unconditionally stable, but they have some problems with approximation. 2 The schemes are conditionally convergent and have an additional term Oðs2 h Þ in the truncation error.
q
This work was supported by the Russian Foundation for Basic Research (project 14-01-00785).
⇑ Address: Nuclear Safety Institute, Russian Academy of Sciences, 52, B. Tulskaya, 115191 Moscow, Russia. E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2014.10.124 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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First explicit difference schemes with traveling computations for parabolic equations of second order were proposed by Saul’yev in the book [8] (the book in Russian was published in 1960). In view of explicit–implicit inhomogeneity of approximation in time, the author called them by asymmetric schemes. Further fundamental result was obtained by Samarskii in the work [9], where these schemes were treated as factorized operator-difference schemes with the additive splitting of the problem operator (matrix) into two terms that are adjoint to each other. Considering systems of ordinary differential equations, we split the original matrix into the lower and upper triangular matrices, i.e., we speak of the alternating triangle method (ATM). In solving steady-state problems on the basis of such the operator splitting approach, we obtain iterative alternating triangle method [10] and the explicit alternating direction schemes [11]. Further applications of explicit schemes with traveling computations for solving parabolic BVPs can be attributed to the works performed by Evans with co-authors [12,13]. Taking into account peculiarities of computations, there are highlighted explicit schemes of the Group Explicit (Alternating Group Explicit) method. Possibilities of explicit schemes under consideration for solving BVPs for parabolic equations on parallel computers are actively discussed in the literature (see, e.g., [14,15]). Explicit schemes with traveling computations are also used for time-dependent convection–diffusion problems [16,17]. In this paper, we propose a multilevel modification of the alternating triangle method (MLATM). To improve the accuracy of ATM schemes, we add a corrective term with the time derivative, which is taken from the previous time level. The original two-level scheme becomes a three-level scheme, but it preserve stability properties (the MLATM scheme is unconditionally stable). Because of this, the truncation error is reduced by an order of the time step magnitude: for the second-order para2 bolic equation, the additional term in the truncation error is Oðs3 h Þ. The stability is studied on the basis of the stability (well-posedness) theory for operator-difference schemes in finite-dimensional Hilbert spaces [6,7,18]. The paper is organized as follows. In Section 2, we consider a model problem in a rectangle for a parabolic equation of second order. Stability conditions are also formulated here for the explicit scheme. Construction and investigation of ATM schemes is performed in Section 3. Section 4 is the core of our work. It describes a modification of the ATM scheme based on the transition from the two-level scheme to a three-level one. Problems for hyperbolic equations of second order are discussed in Section 5. In these problems, the convergence conditions of explicit schemes are acceptable if we apply the standard version of the alternating triangular method. 2. Model problem As a typical example, we study the boundary value problem for a parabolic equation of second order. Let us consider a model two-dimensional parabolic problem in a rectangle
X ¼ fxjx ¼ ðx1 ; x2 Þ; 0 < xa < la ; a ¼ 1; 2g: An unknown function uðx; tÞ satisfies the equation
2 @u X @ @u ¼ f ðx; tÞ; kðxÞ @t a¼1 @xa @xa
x 2 X;
0 < t 6 T;
ð1Þ
where k 6 kðxÞ 6 k; x 2 X, k > 0. The Eq. (1) is supplemented with homogeneous Dirichlet boundary conditions
uðx; tÞ ¼ 0;
x 2 @ X;
0 < t 6 T:
ð2Þ
In addition, we specify the initial condition
uðx; 0Þ ¼ u0 ðxÞ;
x 2 X:
ð3Þ
In X, we define a uniform rectangular grid:
¼ fxjx ¼ ðx1 ; x2 Þ; xa ¼ ia ha ; ia ¼ 0; 1; . . . ; Na ; Na ha ¼ la ; a ¼ 1; 2g x ¼ x [ @ x). For grid functions yðxÞ ¼ 0; x 2 @ x, in the standard way, we introduce a and let x be the set of interior points (x finite-dimensional Hilbert space H ¼ L2 ðxÞ equipped with the scalar product and norm
ðy; wÞ
X yðxÞwðxÞh1 h2 ;
kyk ðy; yÞ1=2 :
x2x
For a positive definite self-adjoint operator D ðD ¼ D > 0Þ, we define the space HD , where
ðy; wÞD ðDy; wÞ;
kykD ðy; yÞD1=2 :
Let us consider a grid operator
A ¼ D1 þ D2 : For one-dimensional grid operators Da : H ! H; a ¼ 1; 2, we have
ðD1 yÞðxÞ ¼
1 yðx1 þ h1 ; x2 Þ yðxÞ yðxÞ yðx1 h1 ; x2 Þ ; kðx1 þ 0:5h1 ; x2 Þ kðx1 0:5h1 ; x2 Þ h1 h1 h1
x 2 x;
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P.N. Vabishchevich / Applied Mathematics and Computation 250 (2015) 424–431
ðD2 yÞðxÞ ¼
1 yðx1 ; x2 þ h2 Þ yðxÞ yðxÞ yðx1 ; x2 h2 Þ ; kðx1 ; x2 þ 0:5h2 Þ kðx1 ; x2 0:5h2 Þ h2 h2 h2
x 2 x:
In the class of sufficiently smooth coefficients k and functions u, these operators approximate the differential operators with the second order. In addition [6,10], we have in the space H of grid functions:
Da ¼ Da ; da ¼
4 2
ha
kda E 6 Da 6 kDa E; 2
sin
pha 2la
Da ¼
;
4 2
ha
cos2
pha 2la
;
a ¼ 1; 2;
where E is the identity operator in H. Thus
A ¼ A ;
kdE 6 A 6 kDE;
d¼
2 X da ;
D¼
a¼1
2 X Da ;
ð4Þ
a¼1
After approximation in space, using for the approximate solutions the same notation as in (1)–(3), we obtain the Cauchy problem for the operator-differential equation
du þ Au ¼ f ðx; tÞ; dt uðx; 0Þ ¼ u0 ðxÞ;
x 2 x;
0 < t 6 T;
ð5Þ
x 2 x:
ð6Þ
To solve numerically the problems 5 and 6, we start our consideration with the simplest explicit two-level scheme. Let s be a step of a uniform grid in time such that yn ¼ yðt n Þ; t n ¼ ns, n ¼ 0; 1; . . . ; N; N s ¼ T. Let us approximate Eq. (5) by the explicit two-level scheme
ynþ1 yn
s
þ Ayn ¼ un ;
n ¼ 0; 1; . . . ; N 1;
ð7Þ
where, e.g., un ¼ f ðx; t n Þ. In view of (6), the operator-difference Eq. (7) is supplemented with the initial condition
y0 ¼ u0 :
ð8Þ 2
2
The truncation error of the difference scheme (7) and (8) is Oðjhj þ sÞ, where jhj ¼
2 h1
þ
2 h2 .
Theorem 1. The explicit difference scheme (7) and (8) is stable for
s 6 ð1 eÞs0 ; s0 ¼
2 kAk
ð9Þ
at any 0 < e < 1, and the finite-difference solution satisfies the estimate
kynþ1 k2A 6 ku0 k2A þ
n sX kuk k2 : 2e k¼0
ð10Þ
Proof. Rewrite the scheme (7) in the form
s ynþ1 yn ynþ1 þ yn E A þA ¼ un : 2 s 2 Multiplying this equation scalarly in H by
2syt ¼ 2ðynþ1 yn Þ; we get
2s
s E A yt ; yt þ ðAynþ1 ; ynþ1 Þ ðAyn ; yn Þ ¼ 2sðun ; yt Þ: 2
Under the restriction (9) on a time step, we have
s
E A P eE: 2 To estimate the right-hand side of (11), we use the inequality
ðun ; yt Þ 6 ekyt k2 þ
1 kun k2 : 4e
ð11Þ
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P.N. Vabishchevich / Applied Mathematics and Computation 250 (2015) 424–431
From (11), we arrive at the following level-wise estimate;
kynþ1 k2A 6 kyn k2A þ
s 2e
kun k2 ;
which implies the required estimate (10).
h
Taking into account (4), for the time step, we have
s < s0 , where, for the above-considered model problem, s0 ¼ Oðjhj2 Þ.
3. Schemes of the alternating triangle method Let us decompose the problem operator A into the sum of two operators:
A ¼ A1 þ A2 :
ð12Þ
Individual operator terms in (12) must make it possible to construct splitting schemes based on explicit calculations. In the alternating triangle method [6,9,10], the original matrix is splitted into the upper and lower matrices, which correspond to the operators adjoint to each other:
A1 ¼ A2 :
ð13Þ
With regard to the problem (5) and (6), we have
ðA1 yÞðxÞ ¼
1 yðx1 þ h1 ; x2 Þ yðxÞ 1 yðx1 ; x2 þ h2 Þ yðxÞ kðx1 þ 0:5h1 ; x2 Þ kðx1 ; x2 þ 0:5h2 Þ ; h1 h1 h2 h2
ðA2 yÞðxÞ ¼ kðx1 0:5h1 ; x2 Þ
yðxÞ yðx1 h1 ; x2 Þ yðxÞ yðx1 ; x2 h2 Þ þ kðx1 ; x2 0:5h2 Þ ; h1 h2
x 2 x;
x 2 x:
Thus, we have splitting of fluxes. To solve the problem (5), (6), (12) and (13), we can use various splitting schemes, where the transition to a new time level is associated with solving subproblems that are described by the individual operators A1 and A2 . For the above two-component splitting (12), it is natural to apply factorized additive schemes [6,19]. In this case, we have
ðE þ rsA1 ÞðE þ rsA2 Þ
ynþ1 yn
s
þ Ayn ¼ un ;
n ¼ 0; 1; . . . ; N 1;
ð14Þ
where r is a weight parameter and un ¼ f ðx; rt nþ1 þ ð1 rÞtn Þ. The value r ¼ 0:5 corresponds to the classical Peaceman– Rachford scheme [20], whereas for r ¼ 1, we obtain an operator analog of the Douglas–Rachford scheme [21]. Theorem 2. The factorized scheme of the alternating triangle method (12)–(14) is unconditionally stable in HA under the restriction r P 0:5 . The following a priori estimate holds:
kynþ1 k2A 6 ku0 k2A þ
n sX
2 k¼0
kuk k2 :
ð15Þ
Proof. The factorized operator
B ¼ ðE þ rsA1 ÞðE þ rsA2 Þ for the splitting (12) and (13) with
B ¼ B ¼ E þ rsA þ r
r P 0 is self-adjoint and positive definite. More precisely, we have
2 2
s A1 A2 P E þ rsA:
In the above notation, the scheme (14) can be written as
s ynþ1 yn ynþ1 þ yn B A þA ¼ un : 2 s 2 Under the restriction
r P 0:5, we have
s
B A P E: 2 Multiplication of (16) scalarly in H by 2syt yields the equality
2s
s B A yt ; yt þ ðAynþ1 ; ynþ1 Þ ðAyn ; yn Þ ¼ 2sðun ; yt Þ: 2
Under the restriction (9) on the time step, we have
ð16Þ
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P.N. Vabishchevich / Applied Mathematics and Computation 250 (2015) 424–431
s
E A P eE: 2 For the right-hand side, we use the inequality
1 ðun ; yt Þ 6 kyt k2 þ kun k2 4 and obtain
s
kynþ1 k2A 6 kyn k2A þ kun k2 ; 2 which immediately implies the estimate (15).
h
Special attention should be given to the investigation of the accuracy of the alternating triangle method. The accuracy of the approximate solution of (5) and (6) is estimated without considering the truncation error due to approximation in space. The convergence of the factorized scheme of the alternating triangle method (12)–(14) for the problem (5) and (6) is studied in the standard way. The equation for the error zn ¼ yn un has the form
B
znþ1 zn
s
þ Azn ¼ wn ;
n ¼ 0; 1; . . . ; N 1;
with the truncation error wn . By Theorem 2, the error satisfies estimate
kznþ1 k2A 6
n sX
2 k¼0
kwk k2 :
The truncation error has the form
wn ¼ wnr þ wns ;
ð17Þ
where
wnr ¼
1 2
2
d u
ðt nþ1=2 Þ þ Oðs2 Þ; 2 dt du nþ1=2 ðt Þ þ Oðs3 Þ: wns ¼ r2 s2 A1 A2 dt
r
s
ð18Þ
The first part of the truncation error wnr is standard for the conventional scheme with weights:
ynþ1 yn
s
þ Aðrynþ1 þ ð1 rÞyn Þ ¼ un ;
n ¼ 0; 1; . . . ; N 1;
which converges in HA with the second order with respect to s for r ¼ 0:5, and only with the first order if r – 0:5. In considering the truncation error for explicit schemes of the alternating triangle method, emphasis is on the second part wns in (17) and (18). Taking into account the explicit representation for the operators A1 and A2 in the model problem (5) and (6), we get wns ¼ Oðs2 jhj2 Þ. Because of this, the operator-difference scheme (12)–(14) for the problem (5) and (6) has accuracy
kznþ1 kA 6 M
r
1 s þ s2 þ s2 jhj2 : 2
ð19Þ
This conditionally convergent scheme has strong enough restrictions on a time step. That is why it seems reasonable to modify this scheme of the alternating triangle method (12)–(14) in order to improve accuracy by reducing error wns . 4. Multilevel alternating triangle method The scheme of alternating triangle method (14) is a two-level scheme. We construct a three-level modification of this scheme, which preserves the unconditional stability but demonstrates more acceptable estimates for accuracy. Such schemes are called here as schemes of MLATM (Multi-Level Alternating Triangle Method). Rewrite the scheme (14) as
ðE þ rsAÞ
ynþ1 yn
s
þ r2 s2 A1 A2
ynþ1 yn
s
þ Ayn ¼ un :
Here we have separated the term that corresponds to the standard scheme with weights from the term proportional to which is associated with splitting. For this, we replace the term associated with splitting by
s2 ,
P.N. Vabishchevich / Applied Mathematics and Computation 250 (2015) 424–431
r2 s2 A1 A2
ynþ1 yn
r2 s2 A1 A2
s
yn yn1
s
¼ r2 s3 A1 A2
ynþ1 2yn þ yn1
s2
429
:
After this modification the MLATM scheme takes the form
ðE þ rsAÞ
ynþ1 yn
s
þ r2 s3 A1 A2
ynþ1 2yn þ yn1
s2
þ Ayn ¼ un :
ð20Þ
As in the standard ATM scheme (14), the transition to a new time level in (20) involves the solution of the problem
ðE þ rsA1 ÞðE þ rsA2 Þynþ1 ¼ nn : For the truncation error, now we have the representation (17), where
wnr ¼
1 2
2
d u
ðt nþ1=2 Þ þ Oðs2 Þ; 2 dt 2 d u wns ¼ r2 s3 A1 A2 2 ðtnþ1=2 Þ þ Oðs4 Þ: dt
r
s
ð21Þ
Thus, the error associated with splitting wns decreases by an order of s. If we use the MLATM scheme for the splitting (12) and (13) for the approximate solution of the problem (1)–(3) (explicit schemes), then the truncation error is wns ¼ Oðs3 jhj2 Þ. Our main result is the following. Theorem 3. The scheme of the multilevel alternating triangle method (12), (13) and (21) is unconditionally stable under the restriction r P 0:5. The following a priori estimate holds:
s
E nþ1 6 E n þ kun k2ðEþrsAÞ1 ; 2
ð22Þ
where
nþ1 2 nþ1 2 y þ yn yn þ y E nþ1 ¼ 2 s s A
2
2
:
Eþr2 s3 A1 A2 þs4 ð2r1ÞA
Proof. Taking into account that
ynþ1 yn
s
¼
ynþ1 yn1 s ynþ1 2yn þ yn1 þ ; 2s 2 s2
we write the scheme (20) in the form
C
ynþ1 yn1 ynþ1 2yn þ yn1 þG þ Ayn ¼ un ; 2s s2
ð23Þ
where
C ¼ E þ rsA; G¼
s 2
ðE þ rsAÞ þ r2 s3 A1 A2 :
By
yn ¼
1 nþ1 1 ðy þ 2yn þ yn1 Þ ðynþ1 2yn þ yn1 Þ; 4 4
we can rewrite (23) as
C
ynþ1 yn1 s2 ynþ1 2yn þ yn1 ynþ1 þ 2yn þ yn1 þ G A þA ¼ un : 2s 4 s2 4
ð24Þ
Let
vn ¼
1 n ðy þ yn1 Þ; 2
wn ¼
yn yn1
s
;
then (24) can be written in the form
C
wnþ1 þ wn wnþ1 wn 1 þR þ Aðv nþ1 þ v n Þ ¼ un ; 2 2 s
ð25Þ
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P.N. Vabishchevich / Applied Mathematics and Computation 250 (2015) 424–431
where
R¼G
s2 4
A:
Multiplying scalarly both sides of (25) by
2ðv nþ1 v n Þ ¼ sðwnþ1 þ wn Þ; we get the equality
s 2
ðCðwnþ1 þ wn Þ; wnþ1 þ wn Þ þ ðRðwnþ1 wn Þ; wnþ1 þ wn Þ þ ðAðv nþ1 þ v n Þ; v nþ1 v n Þ ¼ sðun ; wnþ1 þ wn Þ:
ð26Þ
To estimate the right-hand side, we use the inequality
1 1 ðCðwnþ1 þ wn Þ; ðwnþ1 þ wn ÞÞ þ ðC 1 un ; un Þ: 2 2
ðun ; wnþ1 þ wn Þ 6
This makes it possible to get from (26) the inequality
s
E nþ1 6 E n þ ðC 1 un ; un Þ; 2
ð27Þ
where we use the notation
E n ¼ ðAv n ; v n Þ þ ðRwn ; wn Þ: The inequality (27) is the desired a priori estimate, if we show that E n defines the squared norm of the difference solution. By the positive definiteness of A, it is sufficient to require a positiveness of the operator R. In the above notation, we have
R¼
s 2
ðE þ rsAÞ þ r2 s3 A1 A2
Thus, R > 0 for
s2 4
A>
s2 4
ð2r 1ÞA:
r P 0:5. This concludes the proof.
h
5. Hyperbolic equations Special attention should be given to the problem of constructing explicit schemes of the alternating triangle method for hyperbolic equations of second order. As a model problem, we will consider the boundary value problem in a rectangle X for the equation
2 @2u X @ @u ¼ f ðx; tÞ; kðxÞ @xa @t2 a¼1 @xa
x 2 X;
0 < t 6 T:
ð28Þ
The Eq. (28) is supplemented with the boundary condition (2) and two initial conditions:
uðx; 0Þ ¼ u0 ðxÞ;
@u ðx; 0Þ ¼ v 0 ðxÞ; @t
x 2 X:
ð29Þ
After approximation in space (see (5) and (6)), from the problem (2), (28) and (29), we arrive at the problem 2
d u dt
2
x 2 x;
þ Au ¼ f ðx; tÞ;
uðx; 0Þ ¼ u0 ðxÞ;
0 < t 6 T;
du ðx; 0Þ ¼ v 0 ðxÞ; dt
x 2 x:
ð30Þ
ð31Þ
For the operator A, the splitting (12) and (13) takes place. The scheme of the alternating triangle method for the problem (12), (13), (30) and (31) is written [19] like this:
G
ynþ1 2yn þ yn1
s2
þ Ayn ¼ un ;
n ¼ 1; 2; . . . ; N 1;
ð32Þ
where y0 ; y1 are prescribed. The factorized operator G has the form
G ¼ ðE þ rs2 A1 ÞðE þ rs2 A2 Þ: For the scheme (32) and (33), the truncation error associated with splitting is 2
wns ¼ r2 s4 A1 A2
d u dt
2
ðt n Þ þ Oðs5 Þ:
ð33Þ
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431
In the numerically solving problem (2), (28) and (29), the explicit scheme (32) and (33) has the truncation error wns ¼ Oðs4 jhj2 Þ. Such the truncation error is appropriate for many applied problems. This allows us to restrict ourselves to the classical version of explicit schemes for the alternating triangle method without the multilevel modification. It remains to obtain the condition for stability of the scheme (32) and (33). In the above notation, the scheme (32) and (33) can be written as
R
wnþ1 wn
s
1 þ Aðv nþ1 þ v n Þ ¼ un : 2
ð34Þ
In our case, we have
R¼Eþ
r
1 4
s2 A þ r2 s4 A1 A2 P E
ð35Þ
under the restriction r P 0:25. Similarly to (26) and (27), from (34), we get
E nþ1 ¼ E n þ sðun ; wnþ1 þ wn Þ:
ð36Þ
For the right-hand side of (36), we apply the estimates
s s sðun ; wn Þ 6 kun k2R1 þ kwn k2R ; 2
sðun ; wnþ1 Þ 6 Besides, for all
2
s
s
2
2
1þ
kun k2R1 þ
s 2
1þ
s1 2
kwnþ1 k2R :
e > 0, we have
1 þ es < expðesÞ: In view of these estimates, from (36), it follows the level-wise estimate
E nþ1 6 expðsÞE n þ expð0:75sÞskun k2R1 ;
ð37Þ
which ensures the stability of the solution with respect to the initial data and the right-hand side. This proves the following statement. Theorem 4. The scheme of the alternating triangle method (12), (13), (32) and (33) is unconditionally stable under the restriction r P 0:25. The solution satisfies the estimate (35) and (37), where
nþ1 2 nþ1 2 y þ yn yn þ y : E nþ1 ¼ 2 s A
R
References [1] L. Angermann, P. Knabner, Numerical Methods for Elliptic and Parabolic Partial Differential Equations, Springer Verlag, 2003. [2] U.M. Ascher, Numerical Methods for Evolutionary Differential Equations, Society for Industrial Mathematics, 2008. [3] R.J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations. Steady-state and Time-dependent Problems, Society for Industrial Mathematics, 2007. [4] G.I. Marchuk, Splitting and alternating direction methods, in: J.-L. Lions, P.G. Ciarlet (Eds.), Handbook of Numerical Analysis, vol. I, North-Holland, 1990, pp. 197–462. [5] N.N. Yanenko, The Method of Fractional Steps. The Solution of Problems of Mathematical Physics in Several Variables, Springer, 1971. [6] A.A. Samarskii, The Theory of Difference Schemes, Marcel Dekker, New York, 2001. [7] A.A. Samarskii, P.P. Matus, P.N. Vabishchevich, Difference Schemes with Operator Factors, Kluwer Academic Pub, 2002. [8] V.K. Saul’ev, Integration of Equations of Parabolic Type, Pergamon Press, 1964. [9] A.A. Samarskii, An economical algorithm for the numerical solution of differential and algebraic equations, Zh. Vychisl. Mat. Mat. Fiz. 4 (3) (1964) 580585. in Russian. [10] A.A. Samarskii, E.S. Nikolaev, Numerical Methods for Grid Equations, vol. I, II, Birkhauser Verlag, Basel, 1989. [11] V.P. Il’in, On the explicit alternating direction schemes, lzv. Sib. Otd. Acad. Sci. USSR Ser. Tekhn. Nauk 13 (3) (1967) 97–104. in Russian. [12] D.J. Evans, Alternating group explicit method for the diffusion equation, Appl. Math. Model. 9 (3) (1985) 201–206. [13] D.J. Evans, A.R.B. Abdullah, Group explicit methods for parabolic equations, Int. J. Comput. Math. 14 (1) (1983) 73–105. [14] B. Zhang, W. Li, Age method with variable coefficients for parallel computing, Parallel Algorithms Appl. 5 (3–4) (1995) 219–228. [15] Y. Zhuang, An alternating explicit–implicit domain decomposition method for the parallel solution of parabolic equations, J. Comput. Appl. Math. 206 (1) (2007) 549–566. [16] Q. Feng, B. Zheng, Parallel alternating group explicit iterative method for convection–diffusion equations, in: WSEAS International Conference. Proceedings. Mathematics and Computers in Science and Engineering, World Scientific and Engineering Academy and Society, 2009, pp. 383–387. no. 8. [17] W.-Q. Wang, The alternating segment Crank–Nicolson method for solving convection–diffusion equation with variable coefficient, Appl. Math. Mech. 24 (2003) 32–42. [18] A.A. Samarskii, A.V. Gulin, Stability of Difference Schemes, Nauka, Moscow, 1973. in Russian. [19] P.N. Vabishchevich, Additive Operator-Difference Schemes, Splitting Schemes, Walter de Gruyter GmbH, Berlin/Boston, 2013. [20] D.W. Peaceman, H.H. Rachford, The numerical solution of parabolic and elliptic differential equations, J. SIAM 3 (1955) 2841. [21] J.J. Douglas, H.H. Rachford, On the numerical solution of heat conduction problems in two and three space variables, Trans. Am. Math. Soc. 82 (1956) 421439.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Explicit schemes for parabolic and hyperbolic equations q P.N. Vabishchevich ⇑ Nuclear Safety Institute, Russian Academy of Sciences, 52, B. Tulskaya, 115191 Moscow, Russia North-Eastern Federal University, 58, Belinskogo, 677000 Yakutsk, Russia
a r t i c l e
i n f o
Keywords: Parabolic equation Hyperbolic equation Finite difference schemes Explicit schemes Alternating triangle method
a b s t r a c t Standard explicit schemes for parabolic equations are not very convenient for computing practice due to the fact that they have strong restrictions on a time step. More promising explicit schemes are associated with explicit–implicit splitting of the problem operator (Saul’yev asymmetric schemes, explicit alternating direction (ADE) schemes, group explicit method). These schemes belong to the class of unconditionally stable schemes, but they demonstrate bad approximation properties. These explicit schemes are treated as schemes of the alternating triangle method and can be considered as factorized schemes where the problem operator is splitted into the sum of two operators that are adjoint to each other. Here we propose a multilevel modification of the alternating triangle method, which demonstrates better properties in terms of accuracy. We also consider explicit schemes of the alternating triangle method for the numerical solution of boundary value problems for hyperbolic equations of second order. The study is based on the general theory of stability (well-posedness) for operator-difference schemes. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction In the numerical solution of boundary value problems for evolutionary equations, emphasis is on the approximation in time [1–3]. For parabolic equations of second order, unconditionally stable schemes are based on implicit approximations. In this case, we must solve the corresponding boundary value problem for an elliptic equation at every new time level. To reduce computational costs, explicit schemes or different variants of operator-splitting schemes are employed [4,5]. Explicit schemes have evident advantages over implicit schemes in terms of computational implementation. This advantage is especially pronounced in the construction of computational algorithms oriented to parallel computing systems. At the same time explicit schemes have the well-known disadvantage that is associated with strong restrictions on an admis2 sible time step. For parabolic equations, the stability restriction has the form s < s0 ¼ Oðh Þ, where s is the time step and h is the step of the spatial grid [6,7]. Some promises are connected with explicit schemes, where calculations are organized in the form of traveling computations. In fact, such schemes are based on the decomposition of the problem operator into two operators, where only one of them is referred to a new time level. That is why such schemes with inhomogeneous approximation in time are called explicit–implicit schemes. These schemes are unconditionally stable, but they have some problems with approximation. 2 The schemes are conditionally convergent and have an additional term Oðs2 h Þ in the truncation error.
q
This work was supported by the Russian Foundation for Basic Research (project 14-01-00785).
⇑ Address: Nuclear Safety Institute, Russian Academy of Sciences, 52, B. Tulskaya, 115191 Moscow, Russia. E-mail address: [email protected] http://dx.doi.org/10.1016/j.amc.2014.10.124 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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First explicit difference schemes with traveling computations for parabolic equations of second order were proposed by Saul’yev in the book [8] (the book in Russian was published in 1960). In view of explicit–implicit inhomogeneity of approximation in time, the author called them by asymmetric schemes. Further fundamental result was obtained by Samarskii in the work [9], where these schemes were treated as factorized operator-difference schemes with the additive splitting of the problem operator (matrix) into two terms that are adjoint to each other. Considering systems of ordinary differential equations, we split the original matrix into the lower and upper triangular matrices, i.e., we speak of the alternating triangle method (ATM). In solving steady-state problems on the basis of such the operator splitting approach, we obtain iterative alternating triangle method [10] and the explicit alternating direction schemes [11]. Further applications of explicit schemes with traveling computations for solving parabolic BVPs can be attributed to the works performed by Evans with co-authors [12,13]. Taking into account peculiarities of computations, there are highlighted explicit schemes of the Group Explicit (Alternating Group Explicit) method. Possibilities of explicit schemes under consideration for solving BVPs for parabolic equations on parallel computers are actively discussed in the literature (see, e.g., [14,15]). Explicit schemes with traveling computations are also used for time-dependent convection–diffusion problems [16,17]. In this paper, we propose a multilevel modification of the alternating triangle method (MLATM). To improve the accuracy of ATM schemes, we add a corrective term with the time derivative, which is taken from the previous time level. The original two-level scheme becomes a three-level scheme, but it preserve stability properties (the MLATM scheme is unconditionally stable). Because of this, the truncation error is reduced by an order of the time step magnitude: for the second-order para2 bolic equation, the additional term in the truncation error is Oðs3 h Þ. The stability is studied on the basis of the stability (well-posedness) theory for operator-difference schemes in finite-dimensional Hilbert spaces [6,7,18]. The paper is organized as follows. In Section 2, we consider a model problem in a rectangle for a parabolic equation of second order. Stability conditions are also formulated here for the explicit scheme. Construction and investigation of ATM schemes is performed in Section 3. Section 4 is the core of our work. It describes a modification of the ATM scheme based on the transition from the two-level scheme to a three-level one. Problems for hyperbolic equations of second order are discussed in Section 5. In these problems, the convergence conditions of explicit schemes are acceptable if we apply the standard version of the alternating triangular method. 2. Model problem As a typical example, we study the boundary value problem for a parabolic equation of second order. Let us consider a model two-dimensional parabolic problem in a rectangle
X ¼ fxjx ¼ ðx1 ; x2 Þ; 0 < xa < la ; a ¼ 1; 2g: An unknown function uðx; tÞ satisfies the equation
2 @u X @ @u ¼ f ðx; tÞ; kðxÞ @t a¼1 @xa @xa
x 2 X;
0 < t 6 T;
ð1Þ
where k 6 kðxÞ 6 k; x 2 X, k > 0. The Eq. (1) is supplemented with homogeneous Dirichlet boundary conditions
uðx; tÞ ¼ 0;
x 2 @ X;
0 < t 6 T:
ð2Þ
In addition, we specify the initial condition
uðx; 0Þ ¼ u0 ðxÞ;
x 2 X:
ð3Þ
In X, we define a uniform rectangular grid:
¼ fxjx ¼ ðx1 ; x2 Þ; xa ¼ ia ha ; ia ¼ 0; 1; . . . ; Na ; Na ha ¼ la ; a ¼ 1; 2g x ¼ x [ @ x). For grid functions yðxÞ ¼ 0; x 2 @ x, in the standard way, we introduce a and let x be the set of interior points (x finite-dimensional Hilbert space H ¼ L2 ðxÞ equipped with the scalar product and norm
ðy; wÞ
X yðxÞwðxÞh1 h2 ;
kyk ðy; yÞ1=2 :
x2x
For a positive definite self-adjoint operator D ðD ¼ D > 0Þ, we define the space HD , where
ðy; wÞD ðDy; wÞ;
kykD ðy; yÞD1=2 :
Let us consider a grid operator
A ¼ D1 þ D2 : For one-dimensional grid operators Da : H ! H; a ¼ 1; 2, we have
ðD1 yÞðxÞ ¼
1 yðx1 þ h1 ; x2 Þ yðxÞ yðxÞ yðx1 h1 ; x2 Þ ; kðx1 þ 0:5h1 ; x2 Þ kðx1 0:5h1 ; x2 Þ h1 h1 h1
x 2 x;
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P.N. Vabishchevich / Applied Mathematics and Computation 250 (2015) 424–431
ðD2 yÞðxÞ ¼
1 yðx1 ; x2 þ h2 Þ yðxÞ yðxÞ yðx1 ; x2 h2 Þ ; kðx1 ; x2 þ 0:5h2 Þ kðx1 ; x2 0:5h2 Þ h2 h2 h2
x 2 x:
In the class of sufficiently smooth coefficients k and functions u, these operators approximate the differential operators with the second order. In addition [6,10], we have in the space H of grid functions:
Da ¼ Da ; da ¼
4 2
ha
kda E 6 Da 6 kDa E; 2
sin
pha 2la
Da ¼
;
4 2
ha
cos2
pha 2la
;
a ¼ 1; 2;
where E is the identity operator in H. Thus
A ¼ A ;
kdE 6 A 6 kDE;
d¼
2 X da ;
D¼
a¼1
2 X Da ;
ð4Þ
a¼1
After approximation in space, using for the approximate solutions the same notation as in (1)–(3), we obtain the Cauchy problem for the operator-differential equation
du þ Au ¼ f ðx; tÞ; dt uðx; 0Þ ¼ u0 ðxÞ;
x 2 x;
0 < t 6 T;
ð5Þ
x 2 x:
ð6Þ
To solve numerically the problems 5 and 6, we start our consideration with the simplest explicit two-level scheme. Let s be a step of a uniform grid in time such that yn ¼ yðt n Þ; t n ¼ ns, n ¼ 0; 1; . . . ; N; N s ¼ T. Let us approximate Eq. (5) by the explicit two-level scheme
ynþ1 yn
s
þ Ayn ¼ un ;
n ¼ 0; 1; . . . ; N 1;
ð7Þ
where, e.g., un ¼ f ðx; t n Þ. In view of (6), the operator-difference Eq. (7) is supplemented with the initial condition
y0 ¼ u0 :
ð8Þ 2
2
The truncation error of the difference scheme (7) and (8) is Oðjhj þ sÞ, where jhj ¼
2 h1
þ
2 h2 .
Theorem 1. The explicit difference scheme (7) and (8) is stable for
s 6 ð1 eÞs0 ; s0 ¼
2 kAk
ð9Þ
at any 0 < e < 1, and the finite-difference solution satisfies the estimate
kynþ1 k2A 6 ku0 k2A þ
n sX kuk k2 : 2e k¼0
ð10Þ
Proof. Rewrite the scheme (7) in the form
s ynþ1 yn ynþ1 þ yn E A þA ¼ un : 2 s 2 Multiplying this equation scalarly in H by
2syt ¼ 2ðynþ1 yn Þ; we get
2s
s E A yt ; yt þ ðAynþ1 ; ynþ1 Þ ðAyn ; yn Þ ¼ 2sðun ; yt Þ: 2
Under the restriction (9) on a time step, we have
s
E A P eE: 2 To estimate the right-hand side of (11), we use the inequality
ðun ; yt Þ 6 ekyt k2 þ
1 kun k2 : 4e
ð11Þ
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From (11), we arrive at the following level-wise estimate;
kynþ1 k2A 6 kyn k2A þ
s 2e
kun k2 ;
which implies the required estimate (10).
h
Taking into account (4), for the time step, we have
s < s0 , where, for the above-considered model problem, s0 ¼ Oðjhj2 Þ.
3. Schemes of the alternating triangle method Let us decompose the problem operator A into the sum of two operators:
A ¼ A1 þ A2 :
ð12Þ
Individual operator terms in (12) must make it possible to construct splitting schemes based on explicit calculations. In the alternating triangle method [6,9,10], the original matrix is splitted into the upper and lower matrices, which correspond to the operators adjoint to each other:
A1 ¼ A2 :
ð13Þ
With regard to the problem (5) and (6), we have
ðA1 yÞðxÞ ¼
1 yðx1 þ h1 ; x2 Þ yðxÞ 1 yðx1 ; x2 þ h2 Þ yðxÞ kðx1 þ 0:5h1 ; x2 Þ kðx1 ; x2 þ 0:5h2 Þ ; h1 h1 h2 h2
ðA2 yÞðxÞ ¼ kðx1 0:5h1 ; x2 Þ
yðxÞ yðx1 h1 ; x2 Þ yðxÞ yðx1 ; x2 h2 Þ þ kðx1 ; x2 0:5h2 Þ ; h1 h2
x 2 x;
x 2 x:
Thus, we have splitting of fluxes. To solve the problem (5), (6), (12) and (13), we can use various splitting schemes, where the transition to a new time level is associated with solving subproblems that are described by the individual operators A1 and A2 . For the above two-component splitting (12), it is natural to apply factorized additive schemes [6,19]. In this case, we have
ðE þ rsA1 ÞðE þ rsA2 Þ
ynþ1 yn
s
þ Ayn ¼ un ;
n ¼ 0; 1; . . . ; N 1;
ð14Þ
where r is a weight parameter and un ¼ f ðx; rt nþ1 þ ð1 rÞtn Þ. The value r ¼ 0:5 corresponds to the classical Peaceman– Rachford scheme [20], whereas for r ¼ 1, we obtain an operator analog of the Douglas–Rachford scheme [21]. Theorem 2. The factorized scheme of the alternating triangle method (12)–(14) is unconditionally stable in HA under the restriction r P 0:5 . The following a priori estimate holds:
kynþ1 k2A 6 ku0 k2A þ
n sX
2 k¼0
kuk k2 :
ð15Þ
Proof. The factorized operator
B ¼ ðE þ rsA1 ÞðE þ rsA2 Þ for the splitting (12) and (13) with
B ¼ B ¼ E þ rsA þ r
r P 0 is self-adjoint and positive definite. More precisely, we have
2 2
s A1 A2 P E þ rsA:
In the above notation, the scheme (14) can be written as
s ynþ1 yn ynþ1 þ yn B A þA ¼ un : 2 s 2 Under the restriction
r P 0:5, we have
s
B A P E: 2 Multiplication of (16) scalarly in H by 2syt yields the equality
2s
s B A yt ; yt þ ðAynþ1 ; ynþ1 Þ ðAyn ; yn Þ ¼ 2sðun ; yt Þ: 2
Under the restriction (9) on the time step, we have
ð16Þ
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P.N. Vabishchevich / Applied Mathematics and Computation 250 (2015) 424–431
s
E A P eE: 2 For the right-hand side, we use the inequality
1 ðun ; yt Þ 6 kyt k2 þ kun k2 4 and obtain
s
kynþ1 k2A 6 kyn k2A þ kun k2 ; 2 which immediately implies the estimate (15).
h
Special attention should be given to the investigation of the accuracy of the alternating triangle method. The accuracy of the approximate solution of (5) and (6) is estimated without considering the truncation error due to approximation in space. The convergence of the factorized scheme of the alternating triangle method (12)–(14) for the problem (5) and (6) is studied in the standard way. The equation for the error zn ¼ yn un has the form
B
znþ1 zn
s
þ Azn ¼ wn ;
n ¼ 0; 1; . . . ; N 1;
with the truncation error wn . By Theorem 2, the error satisfies estimate
kznþ1 k2A 6
n sX
2 k¼0
kwk k2 :
The truncation error has the form
wn ¼ wnr þ wns ;
ð17Þ
where
wnr ¼
1 2
2
d u
ðt nþ1=2 Þ þ Oðs2 Þ; 2 dt du nþ1=2 ðt Þ þ Oðs3 Þ: wns ¼ r2 s2 A1 A2 dt
r
s
ð18Þ
The first part of the truncation error wnr is standard for the conventional scheme with weights:
ynþ1 yn
s
þ Aðrynþ1 þ ð1 rÞyn Þ ¼ un ;
n ¼ 0; 1; . . . ; N 1;
which converges in HA with the second order with respect to s for r ¼ 0:5, and only with the first order if r – 0:5. In considering the truncation error for explicit schemes of the alternating triangle method, emphasis is on the second part wns in (17) and (18). Taking into account the explicit representation for the operators A1 and A2 in the model problem (5) and (6), we get wns ¼ Oðs2 jhj2 Þ. Because of this, the operator-difference scheme (12)–(14) for the problem (5) and (6) has accuracy
kznþ1 kA 6 M
r
1 s þ s2 þ s2 jhj2 : 2
ð19Þ
This conditionally convergent scheme has strong enough restrictions on a time step. That is why it seems reasonable to modify this scheme of the alternating triangle method (12)–(14) in order to improve accuracy by reducing error wns . 4. Multilevel alternating triangle method The scheme of alternating triangle method (14) is a two-level scheme. We construct a three-level modification of this scheme, which preserves the unconditional stability but demonstrates more acceptable estimates for accuracy. Such schemes are called here as schemes of MLATM (Multi-Level Alternating Triangle Method). Rewrite the scheme (14) as
ðE þ rsAÞ
ynþ1 yn
s
þ r2 s2 A1 A2
ynþ1 yn
s
þ Ayn ¼ un :
Here we have separated the term that corresponds to the standard scheme with weights from the term proportional to which is associated with splitting. For this, we replace the term associated with splitting by
s2 ,
P.N. Vabishchevich / Applied Mathematics and Computation 250 (2015) 424–431
r2 s2 A1 A2
ynþ1 yn
r2 s2 A1 A2
s
yn yn1
s
¼ r2 s3 A1 A2
ynþ1 2yn þ yn1
s2
429
:
After this modification the MLATM scheme takes the form
ðE þ rsAÞ
ynþ1 yn
s
þ r2 s3 A1 A2
ynþ1 2yn þ yn1
s2
þ Ayn ¼ un :
ð20Þ
As in the standard ATM scheme (14), the transition to a new time level in (20) involves the solution of the problem
ðE þ rsA1 ÞðE þ rsA2 Þynþ1 ¼ nn : For the truncation error, now we have the representation (17), where
wnr ¼
1 2
2
d u
ðt nþ1=2 Þ þ Oðs2 Þ; 2 dt 2 d u wns ¼ r2 s3 A1 A2 2 ðtnþ1=2 Þ þ Oðs4 Þ: dt
r
s
ð21Þ
Thus, the error associated with splitting wns decreases by an order of s. If we use the MLATM scheme for the splitting (12) and (13) for the approximate solution of the problem (1)–(3) (explicit schemes), then the truncation error is wns ¼ Oðs3 jhj2 Þ. Our main result is the following. Theorem 3. The scheme of the multilevel alternating triangle method (12), (13) and (21) is unconditionally stable under the restriction r P 0:5. The following a priori estimate holds:
s
E nþ1 6 E n þ kun k2ðEþrsAÞ1 ; 2
ð22Þ
where
nþ1 2 nþ1 2 y þ yn yn þ y E nþ1 ¼ 2 s s A
2
2
:
Eþr2 s3 A1 A2 þs4 ð2r1ÞA
Proof. Taking into account that
ynþ1 yn
s
¼
ynþ1 yn1 s ynþ1 2yn þ yn1 þ ; 2s 2 s2
we write the scheme (20) in the form
C
ynþ1 yn1 ynþ1 2yn þ yn1 þG þ Ayn ¼ un ; 2s s2
ð23Þ
where
C ¼ E þ rsA; G¼
s 2
ðE þ rsAÞ þ r2 s3 A1 A2 :
By
yn ¼
1 nþ1 1 ðy þ 2yn þ yn1 Þ ðynþ1 2yn þ yn1 Þ; 4 4
we can rewrite (23) as
C
ynþ1 yn1 s2 ynþ1 2yn þ yn1 ynþ1 þ 2yn þ yn1 þ G A þA ¼ un : 2s 4 s2 4
ð24Þ
Let
vn ¼
1 n ðy þ yn1 Þ; 2
wn ¼
yn yn1
s
;
then (24) can be written in the form
C
wnþ1 þ wn wnþ1 wn 1 þR þ Aðv nþ1 þ v n Þ ¼ un ; 2 2 s
ð25Þ
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P.N. Vabishchevich / Applied Mathematics and Computation 250 (2015) 424–431
where
R¼G
s2 4
A:
Multiplying scalarly both sides of (25) by
2ðv nþ1 v n Þ ¼ sðwnþ1 þ wn Þ; we get the equality
s 2
ðCðwnþ1 þ wn Þ; wnþ1 þ wn Þ þ ðRðwnþ1 wn Þ; wnþ1 þ wn Þ þ ðAðv nþ1 þ v n Þ; v nþ1 v n Þ ¼ sðun ; wnþ1 þ wn Þ:
ð26Þ
To estimate the right-hand side, we use the inequality
1 1 ðCðwnþ1 þ wn Þ; ðwnþ1 þ wn ÞÞ þ ðC 1 un ; un Þ: 2 2
ðun ; wnþ1 þ wn Þ 6
This makes it possible to get from (26) the inequality
s
E nþ1 6 E n þ ðC 1 un ; un Þ; 2
ð27Þ
where we use the notation
E n ¼ ðAv n ; v n Þ þ ðRwn ; wn Þ: The inequality (27) is the desired a priori estimate, if we show that E n defines the squared norm of the difference solution. By the positive definiteness of A, it is sufficient to require a positiveness of the operator R. In the above notation, we have
R¼
s 2
ðE þ rsAÞ þ r2 s3 A1 A2
Thus, R > 0 for
s2 4
A>
s2 4
ð2r 1ÞA:
r P 0:5. This concludes the proof.
h
5. Hyperbolic equations Special attention should be given to the problem of constructing explicit schemes of the alternating triangle method for hyperbolic equations of second order. As a model problem, we will consider the boundary value problem in a rectangle X for the equation
2 @2u X @ @u ¼ f ðx; tÞ; kðxÞ @xa @t2 a¼1 @xa
x 2 X;
0 < t 6 T:
ð28Þ
The Eq. (28) is supplemented with the boundary condition (2) and two initial conditions:
uðx; 0Þ ¼ u0 ðxÞ;
@u ðx; 0Þ ¼ v 0 ðxÞ; @t
x 2 X:
ð29Þ
After approximation in space (see (5) and (6)), from the problem (2), (28) and (29), we arrive at the problem 2
d u dt
2
x 2 x;
þ Au ¼ f ðx; tÞ;
uðx; 0Þ ¼ u0 ðxÞ;
0 < t 6 T;
du ðx; 0Þ ¼ v 0 ðxÞ; dt
x 2 x:
ð30Þ
ð31Þ
For the operator A, the splitting (12) and (13) takes place. The scheme of the alternating triangle method for the problem (12), (13), (30) and (31) is written [19] like this:
G
ynþ1 2yn þ yn1
s2
þ Ayn ¼ un ;
n ¼ 1; 2; . . . ; N 1;
ð32Þ
where y0 ; y1 are prescribed. The factorized operator G has the form
G ¼ ðE þ rs2 A1 ÞðE þ rs2 A2 Þ: For the scheme (32) and (33), the truncation error associated with splitting is 2
wns ¼ r2 s4 A1 A2
d u dt
2
ðt n Þ þ Oðs5 Þ:
ð33Þ
P.N. Vabishchevich / Applied Mathematics and Computation 250 (2015) 424–431
431
In the numerically solving problem (2), (28) and (29), the explicit scheme (32) and (33) has the truncation error wns ¼ Oðs4 jhj2 Þ. Such the truncation error is appropriate for many applied problems. This allows us to restrict ourselves to the classical version of explicit schemes for the alternating triangle method without the multilevel modification. It remains to obtain the condition for stability of the scheme (32) and (33). In the above notation, the scheme (32) and (33) can be written as
R
wnþ1 wn
s
1 þ Aðv nþ1 þ v n Þ ¼ un : 2
ð34Þ
In our case, we have
R¼Eþ
r
1 4
s2 A þ r2 s4 A1 A2 P E
ð35Þ
under the restriction r P 0:25. Similarly to (26) and (27), from (34), we get
E nþ1 ¼ E n þ sðun ; wnþ1 þ wn Þ:
ð36Þ
For the right-hand side of (36), we apply the estimates
s s sðun ; wn Þ 6 kun k2R1 þ kwn k2R ; 2
sðun ; wnþ1 Þ 6 Besides, for all
2
s
s
2
2
1þ
kun k2R1 þ
s 2
1þ
s1 2
kwnþ1 k2R :
e > 0, we have
1 þ es < expðesÞ: In view of these estimates, from (36), it follows the level-wise estimate
E nþ1 6 expðsÞE n þ expð0:75sÞskun k2R1 ;
ð37Þ
which ensures the stability of the solution with respect to the initial data and the right-hand side. This proves the following statement. Theorem 4. The scheme of the alternating triangle method (12), (13), (32) and (33) is unconditionally stable under the restriction r P 0:25. The solution satisfies the estimate (35) and (37), where
nþ1 2 nþ1 2 y þ yn yn þ y : E nþ1 ¼ 2 s A
R
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