Adaptive modified alternating-triangular method Текст научной статьи по специальности «Математика»

UDC 519.6 10.23947/2587-8999-2018-2-1-44-59

Adaptive modified alternating-triangular method* A.I. Sukhinov, A.E. Chistyakov**

Don State Technical University, Rostov-on-Don, Russia

The paper covers the development of adaptive methods for solving convection-diffusion problems. For this class of problems a version of minimal correction adaptive modified alternating-triangular method was developed, as well the convergence rate of the method was analytically estimated. Numerical results have been presented for solving real hydrodynamical problem in shallow water using parallel algorithm of developed method.

Keywords: wave problem; grid equations; adaptive modified alternating triangular iterative

method

Introduction. Often in applied problems, for example, in the mathematical modeling of hydrodynamics, heat and mass transfer, geofiltration, population dynamics and other processes, it becomes necessary to solve equations of the convection-diffusion type. In the case of using implicit schemes and schemes with weights, such problems lead to linear algebraic equations with a nonselfadjoint operator. One approach to solving such problems is the Gaussian symmetrization method [3]. The disadvantage of this method is the quadratic increase in the number of conditionality of the operator of the problem.

In the class of two-layer iterative methods one of the most successful is the alternate-triangular method offered by A.A. Samarskiy is [1]. Later, Academician A.N. Konovalov developed the adaptive version of the ATM [2]. In [4], a modified alternate-triangular iteration method for solving the Dirichlet problem for the Poisson equation is described, as well as a description of the simple iteration method in the case of a nonselfadjoint operator. In [8], a technique for increasing the rate of convergence of ATM with a priori information by improving the spectral estimates of the pre-imposed operator was demonstrated. In this paper, a variant of a modified iterative alternating-triangular method of minimal corrections for the solution of grid equations with a nonselfadjoint operator is developed, and estimates of the rate of convergence are performed. It should also be noted that the proposed version of the method of minimum corrections is a separate method. The developed algorithms were applied to solve the grid equations obtained as a result of approximation of the three-dimensional mathematical model of hydrodynamics of shallow water bodies [9-16]. A parallel implementation of the adaptive modified iterative alternating-triangular method of minimal corrections is described [17-22].

* The research is done with the financial support from the RSF, Projects No. 17-11-01286.

** E-mail: [email protected], [email protected].

Modified alternate-triangular iterative method. In a finite-dimensional Hilbert space H we consider the problem of finding the solution of the operator equation:

Ax = f, A : H ^ H, (1)

where A is a linear, positive definite operator (A > 0). To find the problem (1) we will use an implicit iteration process:

m +1 m

BX—— + Axm = f, B : H ^ H. (2)

t

In equation (2) m is the iteration number, t> 0 is the iteration parameter, and B is the invertible operator. Call of the operator B in (2) must be substantially simpler than the direct inversion of the original operator A in (1). In constructing B we start from the additive representation of the operator A0 is the symmetric part of the operator A .

Ao = R + R2, R = R; (3)

A - A*

Also here and below we use the skew-symmetric part of the operator A : Ax = —-—.

By virtue of (3) (Ay, y) = (Aoy, y) = 2(Ry y) = 2(R2y, y) . Therefore, in (3) R > 0, R2 > 0 . Suppose that in (2)

B = (D + mRx )D-(D + coR2), D = D* > 0, m> 0, y e H, (4)

where D is an operator.

Because A = A * > 0, then B = B * > 0. Relations (2)-(4) specify a modified alternate-triangular method (MATM) for solving the problem if operators R, R and the methods for determining the parameters t , c and operator D .

The variational optimization of MATM. In the nonstationary MATM of the variational

type:

xm+1 — xm

Bx-+ Axm = f (5)

Tm+1

iterative parameters tm+x is chosen for reasons of minimization [1, 4]:

(Dz'"+1, z"'+1) —» min, D = &> 0. (6)

Criterion (6) together with zm+1 = zm — t^BT 1 Azm , gives

0z>" - tih , f)B 1 Az'",z'"~ tih IB 1 Az'") ^ mm (7)

or

{bz\zm)-Tm+l{bB-lAz\zm)-Tm+l{bz\B^

The minimum is reached when

_ (DB 'Az"', z'") + 0zm, B-'Az'") T"'+1~ 2 0Br1Azm,BT1Azm)

Iterative methods of variational type (6), (8), for which D = A*B l A give:

_ (IB1 AB1 Azm, zm) + (IB1 Azm, B-'Azm ) _ Tm+1 = =

_ (AB1 Azm, B-'Azm) + (A B- Azm, B'^Azm) _

" 2(B7rABrrAZ™,ABrrAZ™) =

(AB1 Azm, B^Az7" ) + (AB-lAzm, B "^Azm ) (AB~1Azm, B^Az7" )

2(B AB Azm, AB-lAzm) (B AB Azm, AB~ Azm)

Then the iterative parameters for MATM of minimum corrections are calculated by the formula:

(Awm, wm )

-, Bwm - Axm - f m = 0,1,.... (9)

m+1 (B1 Awm, Awm ), ()

When A = A* > 0, iterative parameters Tm+1 can be calculated according to the method of

(wV")

speedy descent (D = A): rm+l = v ' , m = 0,1,....

(Aw , w )

In [10] was suggested to select a step at each iteration Tm+1, uniformly distributed on the interval

[t2,t], where t is the iterative step, calculated on the basis of the method of early descent t2

minimum corrections (residuals).

The adaptive optimization of MATM minimum corrections. An essential element in this approach is the additional a priori information about the initial problem (1). For MATM, this

information is associated with estimates 8 and A : D < — A, RD_1R2 < A A.

8 4

Let us write down the MATM parameter estimates taking into account the notation introduced::

B = (D + cR )D 1 (D + cR ) = (D - cR )D~1 (D - cR) + 2a A ,

( By, y) = ||( D -cR ) y | ^-1 + 2c ( A y, y) > 2c (Ao y, y).

We obtain an upper bound for the scalar product: (A0y, y) < — (By, y), using a replacement

2c

C0 = B-1/2AB~1/2, we obtain: y2 (C0) = — (10)

2c

We estimate ( Ay, y ) bottom:

B = (D + oR )D \D + oR ) = D + oA + o2RD R <

1 S

—-^(By,y)<(Ay,y) ^ r1(C0) =-

1 o ,

— + o +— A

r 1 2 ^

1 O ,

—+ o+— A £ 4

4),

y

v* 4 y

c 2 V

1 + +— A£ 4

v

(11)

The rate of convergence of the iterative methods depends on the number of conditionality of the operator C0:

KQ) =

4ax (C) ) 2max ( B^ A B ^ )

( Co ) ^ ( B"" A)B"'12 ) ■

(12)

We now note that for the MATM the estimate of the operator (10)2cA - B is unimprovable [2]. Given this fact, we get an estimate of the number of conditionality:

KQ) =

1

2max (B'1/2 A) B 1/2 ) <_

2min ( B112 A) B~m ) " 2o2min ( B1/2 Ao B ^ ) = ^ 2O ( B"1/2 A) B112 y, y )

= max-

( y y )

(By, y) = max-;-r = max

(( D + oA +o2 RDl R2 ) y, y )

y -0 2o( Ao y, y ) y *0 2o( A y, y )

(Dy, y) + o (Ay,y) + o2 (D-lR2y,R2y)

= max-

y *0

2o( AoУ, y )

= max

y * 0

( Dy, y ) 1 o (D lR2y, R2y )

f

2o( A^y, y ) 2 2 ( A,y, y )

= max

y * 0

1

- +

2

yj(Dy, y)(D Ry, R2y )

2 ( Ay, y )

(Dy, y) | ^ KDRy, R2y )

D% y, R2 y ) \ ( Dy^ y )

Its value is minimal when: o =

( Dy, y )

D Ry, R2y ) "

(13)

Thus, we have obtained an estimate for the optimal value of the parameter ro. With allowance for (13), we find the estimate of the condition number:

f

v(C0 ) < max

1

- +

2

V

Dy, y )( DR y, R y )

( A>y, y )

1 f M

< — 1 + 1 —

2 y V hj

24t ' * = A '

The vectory in (13) is defined as follows: = wm, where wm is the correction vector. As an operator we can take the diagonal part of the matrix A . Optimal operator D is constructed from the condition for maximizing the relation — S/A, which determines the number of iterations. In [4] the technique of calculating the operator D for solving thePoisson equation . In [5] this result is generalized to the case of the presence of a linear source function.

The improved iterative MATM. We rewrite the system of difference equations (1) in the form with homogeneous boundary conditions, changing the right-hand side accordingly [4].

Ay =u, Ay = -X(aayx\ + qy, x em (14)

a=1 Xa

y ( x ) = 0, x e^, (15)

u ( x ) = p( x) + px ( x) / h + P ( x ) / h2 + P ( x ) / h2, x em

Where p ( x ) = ^

a ( h, x2, x )'№( 0, x2, x3 ), x = h,

0, 2h < x < l - 2h, a (l, x2, x3 , x2, x3 ), x = l - h,

x2 e m, x ;

2 ( xi , h2 , x3 )'№( X1,0, x3 ) , x2 = K

p ( x ) = ^0, 2h2 < x2 < l2 - 2h2,

2 (xi , l2 , x2 ) ' № (xi, ^2, x3 ) , x2 = l2 — h2 ,

x em, x em ;

P3 (x )=<

a ( x, x2 )'№( x, x ,0), x = h, 0, 2h < x < 4 - 2h, a (x, x2, ^ ) ' № (x, x2,13 ), x3 — h,

X em, X e m •

Source function q, a member of the equation (14) is S -figurative character and in the overwhelming majority of grid nodes is equal to 0.

We represent the scheme of the iterative two-layer modified alternating-triangular method in the form:

v"+1 - v"

(D + mR ) D- (D + mR ) y-+ Ay" = f,

where Ry = ^

aa

— y- +-

v K 2ha

a 1

ax„ 1

y+-qy 6

(16) (17)

a=1

R2 * = -X

3 (at

a„

a-l \

^ +-a

h„ a 2 h

■y--qy 6

(18)

It can be seen that the operators introduced in accordance with equalities (17) and (18) are conjugate on the set of grid functions vanishing at the boundary of the grid: R = R*, A = R + R.

Estimates are obtained for the constants A and 5: SD < R + R, RD R < ^ (R + R )' (19) and net function d(x), determines the elements of the diagonal matrix D.

d ( x ) = X

a-l

a+l + h2y

a »a

2 h„

a

h I

x e a.

We assume that va (x) is the solution of the boundary value problem

(20)

(-qva-

aaxa q 2h 6

, a-1,2,3; XaEffla

(2l)

va- 0, x - 0, x -1 .

a a

We set ca (xß, xs ) - max va (x), x - (x, x2, x3 ) e

a,

(22)

where ca(xß,x)-^

c ( x, x3 ), ( x, x )Eaxa, if a-1; c ( x, x ), ( x, x )eaxa, if a- 2;

c3 (x, x), (X, X , if a = 3.

Let us now consider another set of three-point difference problems:

a n+i

MO,- wa= 0, xa= 0, xa= la, a = 1,2,3;

(23)

Let ba( Xß, x^):

xa e a^, x — ( x1, x^, x3 ) e a.

xß,x8)- maxwa (x), x-(x,x,x3)ea, a -1,2,3,

(24)

where ba (xß,

b (x,x), (x,x)ea2xa3, if a -1; b (x,x), (x,x)eaxa3, if a - 2; (ß,8)-b ( x, x ), ( x, x )eaxa2, if a - 3;

( 2,3) ,if a-1; (l,3) ,if a- 2; (l, 2) ,if a-3.

( o 2 1 oo

then dy ,l < Ay,y 1, 8 = l.

V J V J

We set = max {q ( ^ x2, x3 )}, ( xl, x2, x3 )ea

(25)

y = —max j a (0,X,X),a (k, x,x), max

a ( x, x2, x3 )+a ( x + K xi, x3 )

y2 = ^rmax i a ( x^ x ), a ( ^ x )=

f

max

0< X2 <¡2-^2

a2 ( xi, X2 , X ) ^ a2 ( X, X ^ h2 , X )

M

y = —maxJ a (X,X,0),a (x,X), max

3 h 1 0<X3 <¡3-h,

V yj

f „ V v /"v v v Xi,

a ( , x, ) ^^ a ( x, x, X3 ^^ )

V

A

We introduce constant quantities (coefficients) in accordance with the equalities:

K=ra/(r« +q/3), « =1,2,3.

The expression for the definition of the function d(x) is written as:

f 1 \f 1 \-1

(26)

3

d ( x ) = ]T

hi

+

1 \

f u 2 a axa -1

V k1a 2ha 6

b + c„

'bJ2

V k1a y

(27)

Estimate for the parameter A has the form

I (6* (Xp,x< )^ (^'x< ))2 + (Xp,x< )

A = max

«=1,2,3

xß^mß,xsg°s (

' (ß,^) =

a = 1,2,3, x ^ e o ß, x5 em,

( 2,3), if a = 1, (1,3), if a = 2, (1,2 ) ,if a = 3.

Taking into account equality <5 = 1, we arrive at the equation defining the parameter co0:

2/VA. (28)

In the case of using the Chebyshev acceleration of the iterative process for the number of iterations n0 (e), required to achieve the specified accuracy e the following estimate is:

VA ln

M = , «0 (e) = O UN ln ( 2 H, N0 = max {N1, N2, N3}. (29)

Convergence of MATM minimum corrections. If xm - x = zm is the vector of error, wm = B 1 Azm is the correction vector, then from equations (1), (2) we obtain:

B(a)zm+1 = B(a)zm -rAzm, (30)

this expression can be written in the following form: wm+1 = wm - tB~1 Awm ,

using a replacement vm = B1/2wm,C = B 1/2AB~1/2, we obtain: B1/2vm+1 = B-y2vm -tB lAB-mvm or

m+1

= v" -rCvm =(E-tC)vm.

(31)

a

a

a=1

We estimate v

m+l

m+l

-IKe-tC)vm||- ((0E-tC0) + ((l-0)e-tQ))i

where C = C0 + C, C0 = C0', C =-C'. We use the triangle inequality:

IMI <9

( t >

E--C0 vm +

I 0 0 J

((l-0) e-tq ) !

We estimate the first term on the right-hand side of (32) (32):

(32)

Eco J"

E -- Q I BV2w

0 0 J

Bi/2 -tb" 1/2AIWm, bV2--B-mA Iw'

w

0

0

/■ _ A

= (Binwm, B1'2 wm ) - 2 - (A0wm, wm ) + ^| (Awm, B- Awm ) .

- (AoWm, wm )

Taking into account that the minimum ratio —9 is achieved with 9 (B Aw 'Aw ), we obtain a version of the method of minimum corrections and the estimate is written:

r } 2 / x (Awm, wm) ,

E--C vm =(By2wm,Binwm)-2^--(Awm,wm) +

. 9 0 V ' (B-Awm, Awm )v '

+

( Awm ,wm ) ^ , x ( Awm ,wm )

, VA 7—(Awm,B-1Awm)-(B1/2wm,B1/2wm)^TAA°_:—L

(B 1 Awm,,wm) I v ^ / V / (B-wm,,wm)

-( V

( AB~invm, B-invm )2 , ( Bm AB-l/2vm ,vm )

( vm vm -'—T- ( vm vm ) V '

V ' ' i T)-l A i?-l/^ m A i?-l/2i V ' '

( B- AB-l/2vm, ab ~1/2vm )

\2 f

- ( v

(vm,vm)-

(Co vm,vm) _ (C0vm, C0vm ) -1 l (C0vm, C0vm ) (vm, vm )

l -

( Bm AB-invm ,1/2 AB-l/2vm )

( Co vm,vm )2 ^

As a result we get:

E0Co J'

-. l -

( Covm, vm )2

(C0vm, C0vm )( vm, vm )

(33)

For the second term in (32), taking (C1vm, vm) = 0 we have:

m

m

m

2

(1 -e) E-xCx ) vm|| =((1 -e) vm, (1 -e) vm )-(xC,vm, (1 -e) vm )-

+[rClvm, rCvm ) = (1 - e)2 ( vm, vm )+e2 The substitution of (33), (34) into (32) gives:

fx X

-Cvm, -Cv" K0 0

(34)

m+1 _____

v < 0

x E--C0 m v +

1 0 0 y

((1 -0) E-xC, ) v

= 0

1 -

= 0

(CoVm,vm 2 1,

^vm, CoVm)(v m,vm) 1

( CoVm, v 2 m )

(CoV m , C0V ^ i( vm, vm )

M + J(1 - 0)2 ( vm, vm ) + 02 ^Cxvm, XCv"

vH + ,

(1 -0)2 +02 0-0-^ v

(vm,vm)

For convenience, we introduce the notation:

s =

1 -

i s-y m m\ ( C0v , v )

2 A

, r =

fx x A — Cv ,— Cv 0 1 0 1

y

(C0vm, C0vm )( vm, vm )

( A om Om )0

(m m\ v ,v )

x =

( b1A of1, A om )'

Evaluation \\vmwith allowance for (35) takes the form:

vm+1 <

We introduce the change of variables 0 =

(0s + ^(1 -0)2 +02r ) I |vm || = (0s + 41 - 20+ 02 (1 + r) ) I \v

1 -7

1 + r

f

\\vm+1 <

1 /1 + r- 2 + 27 + (1 -7)2

1 + r

1 + r

v =

1 -7 1 + r- 2 + 27 +1 - 27 + 7

1 + r V 1 + r

(35)

vm+1 <

1 + /r+ 7

1 + r V 1 + r

(36)

m

m

Let us find the optimal parameter q for this we take the derivative of the right-hand side of

(36):

lr+r

l + r V l + r

r

l + r

+ /)(/ + r2 )

- 0.

Optimal r is equal to: r -

s 2y

V(l + r- s2 )

(37)

(38)

l-r.5 + ir+r

in view of (37) and

1 + k \ 1 + y

v

We substitute (38) into (36):

> 0.

\vm+l <

l -

s r

\(l + r- s2 )

l + r

s + '

r+

s r

(l + r- s2 )

l + r

\v"'\ -

l -

s r

^(l + r- s2 ) r(l + r - s2 ) + s2r

l + r ^(l + r)(l + r- s2 )

v -

s - s

r

\(l + r- s2 ) l + r

r

'(l + r- s2 )

v -

s + (l + r- s2 ) r

\(l + r- s2 )

l + r

v -

s + ^r(l + r- s2 )

l + r

Thus, we obtain the estimate of convergence:

P<

s + '\/r(l + r- s2 )

l + r

(39)

Rate of convergence MATM ( vm = B1'2wm, C0 = B-1/2AB~1/2) depends on:

s -

1

l -

(s-y m m\ Qv , v )

2 A

( C0vm, C0vm )( vm, vm )

1

l -

(A m m\2 ( A0w »w )

B- Awm, wm )( Bwm, wm )

We use the inequality xy < (ax + y / a )2/4 . This inequality holds for all a.

s2 < l--

4 ( A, wm, wm )2

-1--

( a ( Aß'1 Awm, wm ) + ( Bwm, wm ) / a ) 4

\2 '

(a (C0vm, Cvm y(Qvm, vm ) + (Bwm, wm ) / a (Awm, wm )) We now note that for the MATM the estimate of the operator (10) 2mA < B is unimprovable

[2].

s2 <l--

4

As a parameter a take 2ab s < 1 -

( a 2a + ( Bwm, wm ) / a ( Awm, wm )) 4

b + -

( Bwm, wm )

\2 •

2mb (Awm, wm)

The rate of convergence of the method depends on (Bwm, wm(2mA0wm, wm). Let us estimate

this expression

( Bwm, wm ) (( £ + aA +a2 RlD1 R2 ) wm, wm )

2a ( Awm, wm )

2a ( Awm, wm )

f (Dwm, wm )

2a( Awm, wm )

Value a is minimal in case a -

( Dwm, wm )

V(D~lR1wm, Rwm ) '

j

Thus, the optimal y in the formula (39) is given by y = wm, wherer wm is the correction vector. We obtained an estimate for the optimal value of the parameter o.

i r.-——:-ta

v = max

y^0

1

— +

2

V

Dy, y)(D Ry, R2y)

(A y, y )

< 1 2

1+f-

v ^y

S

= 2^ ' à

Take b = v12, it is easy to verify that the expression on the right-hand side of (39) is less than unity for ^ < 1 . For ^ and y the following estimates hold:

s < 1 -

v -1

= s„

.1/2,. .-1/2 \2 V +1

(v1/2 +v-1/2 )2 (B-1 Awm, Awm) , (B 1 Awm,Awm)

r=Bawm,Awm)(1 S )<(b 1 Awm,Awm)

(40)

where v is the number of matrix conditioning C0.

( b - awm, awm )

We introduce the notation k = -

r =

k (1 - s2 ).

( b 1 a wm, A wm)'

We take the derivative of s from the right side of (39) with allowance for (41):

(41)

s + (l - s2 y k (k -1) ^ (1 + k)-2syjk (k -1)

. 1 + kM) J s = (1 - k (1 - s2 ))2

The estimate (39), taking (42) into account, is written as:

^ s max ^y(1 + y- s max2 ) ^

+ ks2

■> 0.

(42)

P<

1 + r

V

(43)

We multiply the numerator and denominator of expression (43) by (smaxy + >/y(1 + y- s max2 ) - y) , as a result we get: P ^ s„

r/ s max lr (1 + r- s max2 )

smaxr + J r( 1 + r- smax2 )

v

In view of (40), we obtain: p < ■

v

Vri (1 + r- smax2 ) l + r

>/n (1 + r- smax2 ) -r

yjr\ (1 + r- smax2 ) +r

h (1 + r- smax2 ) -r

—

or p <

* 1 v -1

* 1 v +1

(44)

+1

u * J/(1 + Smax2 ) where v =v i -• (45)

^(l + r- Smax2 )-r

Expression (45) taking into account (41), (42) takes the form: v - v (VT+I +Jk) , (46) where v is the number of matrix conditioning C0.

For monotone circuits [3]: Pe < 2 ( Pe is the grid Peclet number) there is a restriction k < 1. We obtain the estimate of the parameter v*: v* < v(3 + 2>/2 ).

Thus, the algorithm for calculating the grid equations can be written as:

rm - Axm - f, (47)

B(vm )wm - rm, (48)

CO =

m

(.Dw'\wm)

(49)

( Awm, wm )

^ 2 = 1 -7-V , J-r , (50)

m ( b- Awm, Awm )(Bwm, wm )

( B- Awm, Awm )

k =--1-1---(51)

km i J3-1 A m A m\ ' (51)

(B AW , AW )

1 -

0m =

s 2k

m m

V(1 + km )

1 + km (1 - Sm 2 )'

(52)

( A wm, )

r =6 ——---— (53)

m+1 m (B-lAwm, Awm), ( ;

= xm -rm+iwm, (54)

(55)

Conclusion. In this paper, a variant of the modified alternating-triangular iterative method of minimal corrections for solving grid equations with a nonselfadjoint operator is constructed, and estimates of the rate of convergence are performed. It follows from the obtained estimate that the rate of convergence of the iterative methods depends on the number of conditionality of the symmetric part of the operator A. This fact served as the basis for constructing an alternate triangular method that showed its effectiveness for moderate Peclet numbers. We note that the developed version of the

method of minimum corrections (53) coincides with the classical version of this method (9) in the self-adjoint case.

References

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5. Sukhinov A.I. Modifitsirovannyy poperemenno - treugol'nyy metod dlya zadach teplovodnosti i fil'tratsii // Vychislitel'nyye sistemy i algoritmy. - Rostov - na- Donu: Izd-vo RGU, 1984. - pp. 52-59.

6. Chistyakov A.Ye. Trekhmernaya model' dvizheniya vodnoy sredy v Azovskom more s uchetom transporta soley i tepla// Izvestiya YUFU. Tekhnicheskiye nauki. Tematicheskiy vypusk «Aktual'nyye problemy matematicheskogo modelirovaniya». - Taganrog: Izd-vo TTI YUFU, 2009. - №8 (97). - pp 75-82.

7. Chistyakov A.Ye. Teoreticheskiye otsenki uskoreniya i effektivnosti parallel'noy realizatsii PTM skoreyshego spuska// Izvestiya YUFU. Tekhnicheskiye nauki. Tematicheskiy vypusk «Aktual'nyye problemy matematicheskogo modelirovaniya». - Taganrog: Izd-vo TTI YUFU, 2010, №6 (107). - pp. 237-249.

8. Sukhinov A.I., Shinenya A.V. Uluchsheniye otsenki parametra poperemenno-treugol'nogo iteratsionnogo metoda s apriornoy informatsiyey // Izvestiya YUFU. Tekhnicheskiye nauki. Tematicheskiy vypusk «Aktual'nyye problemy matematicheskogo modelirovaniya». - Taganrog: Izd-vo TTI YUFU, 2010, №6 (107). - pp. 7-15.

10. Sukhinov, A.I. Adaptive modified alternating triangular iterative method for solving grid equations with a non-self-adjoint operator / A.I. Sukhinov, A.Ye. Chistyakov // Mathematical Models and Computer Simulations. - 2012. - V.24, №1. - pp. 3-21.

11. Sukhinov, A.I. Development and investigation of the discrete mathematical model for coastal wave process modelling / A.I. Sukhinov, A.Ye. Chistyakov, Ye.F. Timofeyeva, A.V. Shishenya // Mathematical Models and Computer Simulations. - 2012. - V.24, №8. - pp. 32-44.

12. Sukhinov, A.I. Matematicheskoye modelirovaniye transporta nanosov v pribrezhnoy zone melkovodnykh vodoyemov / A.I. Sukhinov, A.Ye. Chistyakov, Ye.A. Protsenko // Mathematical Models and Computer Simulations. - 2013. - V. 25, № 12. - pp. 65-82.

13. Sukhinov, A.I. Mathematical modeling of sediment transport in the coastal zone of shallow reservoirs / A.I. Sukhinov, A.Ye. Chistyakov, Ye.A. Protsenko // Vychislitel'nyye metody i programmirovaniye: novyye vychislitel'nyye tekhnologii. - 2014. - V. 15, № 4. - pp. 610-620.

14. Sukhinov, A.I. Sravneniye vychislitel'nykh effektivnostey yavnoy i neyavnoy skhem dlya resheniya zadachi transporta nanosov v pribrezhnykh vodnykh sistemakh / A.I. Sukhinov, A.Ye.

Chistyakov, Ye.A. Protsenko, S.A. Shreter // Vychislitel'nyye metody i programmirovaniye: novyye vychislitel'nyye tekhnologii. - 2015. - V. 16, № 3. - pp. 328-338.

15. Nikitina, A.V. Primeneniye skhem povyshennogo poryadka tochnosti dlya resheniya zadach biologocheskoy kinetiki na mnogoprotsessornoy vychislitel'noy sisteme / A.V. Nikitina, A.A. Semenyakina, A.Ye. Chistyakov, Ye.A. Protsenko, I.V. Yakovenko // Fundamental'nyye issledovaniya. - 2015. - № 12-3. - pp. 500-504.

16. Sukhinov, A.I. Mathematical modeling of sediment transport in the coastal zone of shallow reservoirs / A.I. Sukhinov, A.E. Chistyakov, E.A. Protsenko // Mathematical Models and Computer Simulations. - 2014. - V. 6, № 4. - pp. 351-363.

17. Nikitina, A.V. Differentsial'no-igrovaya model' predotvrashcheniya zamorov v melkovodnykh vodoyemakh / A.V. Nikitina, M.V. Puchkin, I.S. Semenov, A.I. Sukhinov, G.A. Ugol'nitskiy, A.B. Usov, A.Ye. Chistyakov // Upravleniye bol'shimi sistemami: sbornik trudov, 2015. - № 55. - S. 343-361.18. Sukhinov, A.I. Mathematical modeling of sediment transport in the coastal zone of shallow reservoirs / A.I. Sukhinov, A.E. Chistyakov, E.A. Protsenko // Mathematical Models and Computer Simulations, 2014. - V. 6, Issue 4. - pp. 351-363.

19. Petrov Igor B. Application of grid-characteristic method for numerical solution of deformable solid mechanics dynamical problems // Computational Mathematics and Information Technologies. - 2017. V.1, No 1. - pp. 1-20.

20. Sukhinov A.I., Sidoryakina V.V., Sukhinov A.A. Sufficient convergence conditions for positive solutions of linearized two-dimensional sediment transport problem // Computational Mathematics and Information Technologies. - 2017. V.1, No 1. - pp. 21-35.

21. Nikitina A.V., Semenyakina A.A. Mathematical modeling of eutrophication processes in Azov Sea on supercomputers // Computational Mathematics and Information Technologies. - 2017. V.1, No 1. - pp. 82-101.

22. Chistyakov A.E., Protsenko E.A., Timofeeva E.F. Mathematical modeling of oscillatory processes with a free boundary // Computational Mathematics and Information Technologies. - 2017. V.1, No 1. - pp. 102-112.

Authors:

Sukhinov Alexander Ivanovich, Don State Technical University (1st Gagarin Square, Rostov-on-Don, Russian Federation), Doctor of Science in Physics and Maths, Professor

Chistyakov Alexander Evgenievich, Don State Technical University (1st Gagarin Square, Rostov-on-Don, Russian Federation), Doctor of Science in Physics and Maths, Associate professor

УДК 519.6

Адаптивный модифицированный попеременно-треугольный метод * А.И. Сухинов, А.Е. Чистяков**

Донской государственный технический университет, Ростов -на-Дону Российская Федерация

Работа посвящена разработке адаптивных методов для решения задач конвекции-диффузии. Для данного класса задач построен вариант адаптивного модифицированного попеременно-треугольного метода минимальных поправок и аналитически получены оценки скорости сходимости для данного метода.

Ключевые слова: сеточные уравнения с несамосопряженным оператором; адаптивный попеременно-треугольный метод; параллельные алгоритмы решения сеточных уравнений

Авторы:

Сухинов Александр Иванович, Донской государственный технический университет (344000 Ростов-на-Дону, пл. Гагарина, д. 1), доктор физико-математических наук, профессор

Чистяков Александр Евгеньевич, Донской государственный технический университет (344000 Ростов-на-Дону, пл. Гагарина, д. 1), доктор физико-математических наук, доцент

* Работа выполнена при поддержке РНФ, проект № 17-11-01286.

** E-mail: [email protected]. [email protected].