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THE PROBLEM OF INSTABILITY OF INTERVAL METHODS FOR LARGE INTERVALS OF INTEGRATION AND ITS SOLVING
DOI 10.24411/2072-8735-2018-10182
Dmitry N. Chernoivan,
South Russian state Polytechnic University (NPI), Novocherkassk, Russia, [email protected]
Polina B. Seredina,
South Russian state Polytechnic University (NPI), Novocherkassk, Russia, [email protected]
Keywords: two-sided method, interval method, large gaps, instability, errors, guaranteed accuracy.
The article describes the solution of the problem of fundamental instability of the known interval and bilateral methods for the numerical solution of odes at large intervals of integration. The problem is related to the exponential growth of the error and a very rigid restriction on the length of the integration interval. We consider examples illustrating the instability of the known two-sided methods (the classical interval Moore method of the first order, the two-sided B. S. Dobronets method of arbitrarily high order accuracy) at large integration intervals. The paper deals with two-sided methods of solving the Cauchy problem, effective in the case of large intervals of integration. According to the characteristics of convergence and stability for the case of nonlinear problems, the proposed methods are significantly superior to known analogues with almost equal order of computational cost at the integration step. Relevant theoretical estimates and examples are provided to confirm the effectiveness of the proposed bilateral methods. The developed methods are divided into the following groups. The methods of the first group are based on a posteriori error estimation of a certain real finite-difference method. Qualitative improvement of the convergence and stability of the two-sided method is achieved by using majorant estimates of the norm of grid green functions corresponding to the scheme of the real method used. When using two-sided methods of the second group, it does not matter in principle, what method is the approximate solution, the error of which is subject to a posteriori evaluation. Majorants are also used to find estimates for the norm of the green function, but not the grid one, but the continuous linearized problem. To calculate majorants for green's functions of a linear problem, a corresponding two-sided method is used, which complicates the algorithm of calculations. Therefore, the use of methods of the second group is advisable in cases where the approximate solution is obtained by an unknown numerical method or by experimental measurements.
Information about authors:
Dmitry N. Chernoivan, South-Russian state Polytechnic University (NPI), master, Novocherkassk, Russia Polina B. Seredina, South-Russian state Polytechnic University (NPI), bachelor's degree, Novocherkassk, Russia
Для цитирования:
Черноиван Д.Н., Середина П.Б. Проблема неустойчивости интервальных методов при больших промежутках интегрирования и ее решение // T-Comm: Телекоммуникации и транспорт. 2018. Том 12. №11. С. 97-102.
For citation:
Chernoivan D.N., Seredina P.B. (2018). The problem of instability of interval methods for large intervals of integration and its solving. T-Comm, vol. 12, no.11, pр. 97-102.
T
Introduction
Finding a numerical solution to the Cauchy problem with guaranteed accuracy is of considerable practical interest. Relevant capabilities interval and bilateral methods [1,2].
However, the known interval and bilateral methods are characterized by a very significant drawback: their error (solution bandwidth) grows exponentially depending on the length of the integration interval (0, //■). For any given accuracy s the maximum allowable length of the integration interval is very weak (logarithmically) depending on the value of the integration step hi //™3"(£. h) ~ 0(ln If1), h 0. That is, to increase the length of the integration interval by 2 times, relatively speaking, it is required to reduce the integration step by e~ = 7,4 times regardless of the order of accuracy of the method.
Let us illustrate this with illustrative examples.
Exam pie 1. You want to solve the problem
dxJdt =J{x), i e (0. tF)\flx) = ~x\ jc(0) = 1.
The difference scheme of the classical interval Moore method of the first order has the form [1]: X,.t = X, + FM)(X,)/) + +Fí2>(A)/Í2/2,
where j - the number of integration step; Xj = [xf, Je/] - interval, including the solution of the problem at t = t¡ (x(t¡) e XJ), t¡ = jh; X,, = 1; A - the interval including the solution on the whole interval of integration (0, tF). Since the exact solution of this problem is *{/) = exp(-f), then the interval A can be chosen equal to A = [0,1].
Consider the derived solutions computed from the right side of the equation: dtx/dt = /*'(*(')), A = 0,1, . ..;/"(a*) = -x;/2\x) = x. Then F* '(X) - the stretch f (x), for example, it is reasonable to choose F,n(X) = -X, F,2'(X) = X. As a result, the scheme of the Moore method takes the form: = X, - /¡X, + [0,l]/r/2. It should be noted that grouping the first and second terms in the scheme is generally incorrect, since the presence of a common factor is associated with a specific type of problem. For example, for the right side f{x) = exp(.v): F( '(X) - exp(X). In a detailed form of recording the difference scheme has the form: .*,-+1~= Xj~ - x/h, Xy+1 Xj - Xj'h + /r/2, Hence, the width of the interval solution wj — x/ — Xj satisfies the recurrent equation: Wj+\ = Wj + hwj + hi 2, j- 0,1,... The solution of this equation is: Wj = [(1 + h)1 - 1 ]h/2 % /i/2[éxp(íj) - 1], 0,1,... Thus, in the limit the interval [0,1] interval, the decision certainly goes in tj > ln(l+2//i)/)/ln( \+h)m - Inh, A->0.
Example 2. Consider the Cauchy problem for the system
dxl/dt = x2, dx2/dt =-xx,t e (tí, /,.■); .V](0)= l. jc2(0> = 0.
Its solution x\ = cos(/), x2 = sin(/) belongs to the interval [-1.1].
1) Finite-difference? scheme of the Moore method of the first order:
X, j+i = X, j + hX2 j + [-1,1 l/r/2, X, j,i = X, j - AX, j + [-1,13^/2,7 = 0,1,...
The width of the interv al solution satisfies the equations:
Wyn = wy+ hwi-j + W2J*i = w2j + hwu + h2, j = 0,1,...
Total width it>'= vrij + w^ can be calculated from equations: Wj+i - Wj + hwj + 2h2, j = 0,1,.,. The solution of the recurrent equation is: Wj=> 2/i[(l + h)j-I]« 2/?[exp(r,) - 11, j = 0,1,... Beyond the allowed interval ¡-i,l | the interval solution will also be released when imax « - In/?, h -> 0,
It is important that methods of higher order accuracy k the score remains the same: f|„aii ~ hi// ' - - k ln/r = 0(In /j~l), h —» 0.
Two-Way method [2]. The solution is sought in the form of X, = s, +[-1,1 ]s/2) +[-a,a]s/l) ,i= 1,2;
where a - some unknown constant, S;, s/n, s/21 - Hermitian splines of the third degree. Splines s;, i = 1,2, approximate the solution xit i = 1,2, of the Cauchy problem. They are constructed according to the approximate solution of the problem by some numerical method, for example liunge-Kutta. Splines s/l), s/2), 1= 1,2, approximate solutions uh v„ i — 1,2, two auxiliary Cauchy problems:
dtt/dl = lVn+w, t e (0, /,,), i/(0) = 0, dv/dt = Wv.te (0,1F), v(0) = 0,
where the vector w has unit components and the matrix W consists of elements Wa =dfjdxu i = 1,2; Wy =\dfjdxj\, i ^ ij = 1,2, where f, i = 5,2 - functions of the right parts of the Cauchy problem.
To ensure bilateralism ratings Xf, i = 1,2, value a is selected according to the formula: a = ma.xfi), (/)AFj (/); t e (0, tF), /=1,2).
Functions 0,{/) u determined by means of relations: %{0 = |<p, (jjijl - ds^fdt + dgjdx^ l2V) + dgjdx2s2{2\t),
= dst1 Vdt - ck/0X\S\\t)' ¿&/dx2s2llXtl q=fi (a-) - ds/dt, - dfjdxh dgjdxj = \df/dx\,
i ',/= 1,2,
Taking into account the initial data of the problem/j(.Y) = x2, fix) = -a* |. Therefore,
Wn = W22 = 0, Wx2 = W2X = 1; v,</) = 0, s/2)(0 = 0. <D,£$ = 1^(5,01,
= ds^/dt - s2u\t), = ds2t])/dt - Si0\t), X,= s, +[-a,a]sp}, V/ e (0, tF),i= 1,2. Suppose splines s„ i = 1,2, approximate the solution of the original problem i — 1,2, with order of accuracy k. Fxact solution of auxiliary problem n(t) equally: u{t) - exp(f) - I, i= 1,2, therefore, %(/) = ^(f) = 1. The width of the two-way solution is: Wj= a[exp(^)- 1], ^ = 0,1,...; a = 0(hh), /i 0.
Hence, for the considered two-sided method of arbitrarily high order of accuracy k, the estimate remains the same unsatisfactory: /nw ns ln/rl = - k in/j = f?(ln h~l), h —^ 0.
Excessive reduction of the integration step leads to a very uneven distribution of the local error of the method and a significant influence of rounding errors, leading to a loss of guarantees of the two-sided solution.
Thus, for the known interval and bilateral methods, there is an exponential increase in error and a very rigid restriction on the length of the integration interval.
The authors have developed bilateral methods for solving the Cauchy problem, effective in the case of large integration intervals. According to the characteristics of convergence and stability for the case of nonlinear problems, the proposed methods are significantly superior to the known analogues [1,2] with almost equal order of computational cost at the integration step.
The developed methods are divided into the following groups. The methods of the first group arc based on a posteriori error estimation of a certain real finite-difference method.
Qualitative improvement of convergence and stability of the two-sided method is achieved by using majorant estimates of the
dx/dt =fix), t e (0, tF)\ a'{0) = Xo,
(1)
where x and / - vector-functions: x = (xi.....
f= if\<—ifm)T\ xo - vector of initial conditions: =
We assume that the solution of the Cauchy problem x(t) exists, is unique and belongs to class if] (N > 2), vector-functions of the right parts of the system (I) also have an appropriate degree of smoothness in some neighborhood of the solution path (*(/),/), t e[0, tF], Let 0 = i„ < < ...< t„ = tF~ splitting a segment [0, tp] with parameter h < htt = const.
The proposed method uses a solution decomposition to obtain two-sided estimates x(t) in Taylor's row with a residual member in the form of Lagrange:
\p+Î
U dt" k! di"" ' (/»+1)! Jf ^-D^l-f-l.....m\j-\.2„...
Denote f(x) = ifx/df , k = 1.....p+\; p < N, then the finite-
difference relations of the Taylor series method for the numerical integration of the Cauchy problem (3) can be written as:
(3)
4=*( o).
The error of the approximate solution xj1, j = 1,2,..., is determined by the values of the residuals = x(lj} - Xj",j — 1,2,..., jc«1 = 0, for which, by virtue of (2) and (3), recurrence relations take place:
(4)
*=i
*%/*!+fr»(x{3fy)h^ /(p+I)/, The ratio (4) can be rewritten as:
The solution of difference equations (5) satisfies the following relations:
norm of the grid green functions corresponding to the scheme of the real method used.
When using two-sided methods of the second group, it does not matter which method is the approximate solution, the error of which is subject to a posteriori evaluation. The majorants for the norm of the green's function are also used to find the estimates, but not for the grid, but for the continuous linearized problem. To calculate the majorants for green's functions of the linear problem, the corresponding two-sided method is used, which complicates the calculation algorithm. Therefore, the use of methods of the second group is adv isable in cases where the approximate solution is obtained by an unknown numerical method or by experimental measurements.
1. Description of a two-sided method with a posteriori
error estimation based on green's grid functions
Without limiting generality, we consider the Cauchy problem for an ODE system in an Autonomous form:
Aj.r-ZfWJV/U. JfixHvr/dx,L.....
5fm(x° )
/=I
dx,
V-y
h) jk! +
+fruM^,))h^/(p + l)!, i = 1.....m;j = 1.2.....
J = I>2""*
a=l
where {Uf} - grid green function, Uf- = Uj Uk'\ k = 0....j; j-0,\.....
Matrixes Uj,j = 1.2,,,., can be found by using recurrence relations:
Uj = (E+hjAj.l)Uj.u j= 1,2,...; Ua = £.
(7)
where E — unit matrix of dimension mxm.
It is obvious that matrix {Uf} are approximations of values at grid nodes of the green function U(t,t) Cauchy problems resulting from linearization of the system (1). Matrix function U(t,z) is the solution of the system
dU(t,xYdt =/Mt)Mt,x), t > t; U(t,t) = E, % > 0.
Denote t/,,={[//}0<ksj<n, P = max{||.v,-'1\;j = I.....«}.
||t/„|j=max{||[;/!|l;0<i<;<«}. where ||Vi| =
max{]xy |; i = I.....m}. matrix norm ||-||| determined by
the rule:
IMIlt :: max{|aii|+...-f-ia<J;i= I—w}, .....m.
From (6) follows inequality:
p < 111/,, | |/jp( p, /i ) < cr( Uh)np{ p, A).
(8)
where v(Uh) > ||tA,||, ||f>||< p(p,A),7=l.....n.
According to the theorems of analysis [3j and the rules for calculating interval estimates [ 1,2] of the majorant p(p.A) can be represented as:
LP
m it-l
= l.....4+max ¡1^^)1^/^+1)1^=1.....»j,
(9)
where Z = [-p,p](l,...,l)r, Fq ,{X) - continuous interval expansion of the matrix function {d fiq>/d xfixftu.1.....m, -ve X c R"\
RP+](X) - continuous interval expansion of the vector function fp+l\x), x e X. A, - a priori uniform interval estimates of the values of the vector-function x(t) on the interval 7}. To Refine the values Ay known interval methods can be used [1,2,4-6]. It is assumed that the arguments xf+Z, A1, contained within the scope of the interval extension definition FAX), RP-\(X).
To find the desired two-sided estimates [xj "-5, Xj °+8] the solution of the Cauchy (1) problem in the grid nodes is sufficient to calculate the largest root of the equation S=a(£/;,)Hp(S,A), possessing the property 8 —> 0 by h —» 0, By virtue of (9), as will be shown in paragraph 2, under very general assumptions 5 = 0(h'') by h-> 0.
Calculations of two-sided estimates can also be carried out using an explicit scheme that does not require solving a nonlinear
7TT
equations = a(Ui,)np(8.h). Denote p/ = 1\\, j = 1,...,«, <Jj(U>,) > max (11Uj\11; 0 < k < /}. From the ratio (6) follows inequality
where
«Î
i-1
Statement I. Under the above assumptions, the following error estimate of the two-sided method obtained by the ratio (8) is valid:
ц < 2lf/(\/C„+D]2) = O(lf). при h-> 0,
(10)
Ihl ~ x
i=l 1=1 H
»• = 1.....+
ZH = ["Ц/-ь Им] (1,...,1)г,у = 1.....n.
From the above written inequalities it is obvious that it is
possible to calculate the estimates from above to \x}, j = 1.....n,
with the help of the relevant explicit recursion formulae.
The calculation of |jt/;,|| a brute-force search leads to a relatively large computational cost of the order (tF /h)2 by li 0, tF-+ oo, while for traditional (non-interval) methods of numerical integration of the Cauchy problem [3] the costs grow in a linear way depending on the number of steps: ~t,./h .
Due to the fact that the value estimate [|i/h|[ it is enough to earry out with relative accuracy of the order of one, the complexity of the corresponding calculations was possible to reduce to the order of magnitude tf/h-ln(t/h) при h—> 0, iF —> да by applying the global interval optimization method [8]. Thus, the order of total computational costs for the considered two-sided method is the same as for the usual (non-interval) methods of Cauchy problem integration.
The described method has significantly improved characteristics of convergence and stability in comparison with interval and bilateral methods [1,2,4, 6] (see statement 2 in paragraph 2).
The method [8] uses a similar (8) inequality to obtain a two-sided solution. The difference is that in the method [8] it is required to find the norm estimate of the green's function of the linearized problem <r(i/(i, r)). In the method described above, instead of the value a(U(t. r)) the estimation of the norm of the grid green's function is used u(U/,), due to this, a significant simplification of the calculation algorithm is achieved.
2. Investigation of the convergence
of the two-sided method
We introduce the following domain of Euclidean space R'":
where v = const.
Let 1 < h/hmi„ < x < ж, where /;„„„ — minimum step of partitioning the interval [0, tF\, x = const.
We will assume that AjC: Q.(tF,v), j ~ 1.....n.
On any subset Cl(rF,v) vector-functions of the right parts of the ODF, system (1), their partial derivatives up to order N > p and the corresponding interval extensions are defined, and their norms are uniformly bounded by quantities that depend only on the parameters tF, v, N, m .
The norm of the green's function of the linearized Cauchy problem is uniformly bounded on the set [0,//.]x[0,/,.-]: ||t/(/,r)|h<cr(t/)<«.
where D = 1/C(,: -4lip , by the condition C0 < UIK1*1, h <D(I, where Co, — a nonnegative value that depends only on parameters tp, v, p, m.
Proof,
According to the known convergence theorems for the methods of numerical integration of the Cauchy problem [1-3], under the assumptions made, there is a uniform convergence on the interval [0,//.] the Taylor series method (3) and the fmite-difference method defined by relations (7) so that the relations are satisfied:
||-)|l < Ci(yjr,p,m)h ". j - I.....n, (11)
WUf-mtjM l £ 0 <k<j ¿n, (12)
by h < Ci(\,tF,p,m).
it follows from inequality (11) that, starting with some
h < C4(vj,,.p.m)
|lx/-x(^)J<v/2.y=l.....«. (13)
Suppose at lirst that fi < v/2. Under this condition, taking into account (13) Xj + Z c Q(/f). j — !.....n. therefore, due to inequality
POU) < p W £ X lkj'1-v; + z)\h) jq\; i = \,...,m;j = 1.....n) +
[<H i-,/=i" 11 max \\Rp+,(.Aj )| hf ¡(p + \)!; j = 1.....n\
a fair assessment p(p,A) < C5(p:/i+ If'1), where Q = Q(v,rF ,p,m). From (8) taking into account (12) and the last inequality follows:
M<C6(p3+A"),
(14)
where Q = CJm\\U,\\ < C&tF{cs(U) + C2h) = Cç(v,/F,/?,/#).
Solving inequality (14) with respect to p find the desired rating:
(.i < 2hf'/{\/Q)+DV2), (15)
D = l/Co2 - 4//', by h < C7(v, tF,p,m).
The estimate ( 15) is correct for a non-negative value D, that is, when the condition is met
Co < 0,5A"P'".
(16)
in force (IS) n = 0(h"), by 0. By h < [v/(2C„)]l y'the assumption, rightly made at first, that p < v/2 and so the statement is proved.
Denote by tFn""i(s, h) the limit length of the integration interval corresponding to the specified accuracy values e and the interval split option h.
Statement 2. Let the partial derivatives of the right parts (1) be up to order p, their interval extensions in the region il( w), the green's function of the corresponding linearized problem is imi-
formly hounded, and the methods used to compute the real solution (3) and the grid green's functions (7) converge uniformly on the half-axis
t>0byh< h\(v,p,m), then by h < nun(/h(v(/?.m), z~p) a fair assessment h) > Cv(v.p.m)h '' \
Proof.
By virtue of the assumptions made, the value of Co in (10) can be selected as C(, = Cq(v,p.m)tF. For the correctness of the estimates according to the statement I and the statement condition 2 it is required that the parameter of the integration interval division does not exceed the value of some value depending only on the parameters v.p.m.
Since p < 2//'C0, then to meet the requirement for the accuracy of the solution is sufficient to fulfill the inequality 2ifG)t/.-< e . To fulfill the constraint (16), it is sufficient that the ratio CgtF<, \flh~pl. Of the last two inequalities, there is also a sufficient restriction on the value oftF '■
tF < 1/{2G,) min(G/i^, h^2). (17)
Value /F",ax(£, h) = h~pl2/(2Cq) by h < min(Cw(v.p,m). £2'p) satisfies the restriction (17), which proves the statement.
For comparison, it can be noted that for interval and bilateral methods described in [1,2,4-61, in General, there is a significantly stricter limit on the value ofi^E, h): i/^e, h) = 0(ln If1), h —> 0, (value e fixed),
3. Examples of effective application
of the method in solving applied problems
Example 1. Cauchy problem for the Duffing equation [3] describing nonlinear oscillations of different nature (in this case without energy dissipation):
ify/dr + ¿0+/) = 0. />0.
The considered ease of the absence of energy dissipation is of additional interest as a rigid test in the study of the properties of convergence and stability of interval and bilateral integration methods.
The initial conditions and numerical parameter in the test problem have the following values: y(0) = 0.5, dy/dt(0)~ 0 , X = 0.25.
In normal form, the Cauchy problem has the form:
dx\/dt = x2. dxi/dt = - a( a',+ ay1), / > 0;
X[(0) = 0.5, *:(0)= 0,
wherey = xj. dy/dt =x2, therefore, fl\x) = (as , - .r,+ ay1)) t.
f\x) = (- k( Xi+Xi, - M 1+ 3*f)*2)7',...,
0
-X(l+3.vf)
->.(1+3aV)
0
It follows from the first integral of the system that the solution of the Cauchy problem satisfies the estimates: |.V]{/)| < *,(0)=0.5, [^(Ol <0.3.
Denote p(p,A) = C^r+Ci. then from (8) follows: p < 2a(Uh)nC2/(\+D12), где D = 1-(2ст(Ул)и)2С1С2. For the purpose of comparative analysis of computational qualities of methods, the Cauchy problem was integrated by the considered two-sided method, as well as by the two-stage (improved) interval method [6] and the two-sided method described
in [2]. The results of calculations on a PC of Pentium type (300 MHz) for the case p = 4 are presented in table,1, where the number 1 is a two-stage interval method [6], and the method number 2 is described in paragraph I of the article.
Table 1
If Methode No 1 Methode No 2
Step h Precision E Time of calculations, min. Step h Precision e Time of calculations, m in.
25.6 I0'J 2-Iff4 0.02 10" 2-L0-" 0,02
51.2 1.5-Iff4 310° 1.5 5- Ю"2 2-10° 0.09
60 5-10° 10"" 5 6-10'2 5-10"s 0.12
70 10° 5 -10*5 30 S-10"2 4-10'5 0.13
102.4 Many hours of time, unac-ceptably small integration step and technical limitations associated with the tin ¡ten ess of the computer's discharge grid 5-103 6 105 0.5
204.8 The same 2.5-Ю-" 6-10'5 1.1
The accuracy of the results was determined by the maximum width of the interval solution in the first case and by the values of the value p in the second.
The data of the table show that the proposed method in respect of convergence and stability significantly exceeds the improved interval method [6] and, moreover, the interval Moore methods [1|. A similar conclusion may be made in respect of described in [2] bilateral numerical method of integration of the Cauchy problem for systems of ODE on the basis of a posteriori evaluation.
Example 2. Calculation of transient in a nonlinear electric circuit 18], The ODE system has the form:
dx\/dt = \/L\{- R\X\ - A'j + i7„,sin(co.v4+(p)),
dxj/dt = l/i2(A-3 - R2x2),
dxy'dt = 1 /Ca(X\ - Xig{Xs) - x2),
dxi/dt = 1, t > 0.
This system was solved by the bilateral method described in item 1 and the improved interval method [6| of the 3rd order of accuracy at the following initial data:
x,(0) = 0 J = 1,2,3,4; /?, = 10, R, = 1, i., = 10"',
L2= 10"\C„= 10"4,
Ump 10; £00 = g0( 1 -a«2), go = 2-10'2; a = 5-10 3; co = IOOti, cp = n/2.
By h =s i.53'10 7w , tF - 210 2 the cost of computing lime when finding a solution described in the article by a two-way method with accuracy ex— 1.310 3 to solve the same problem by the Moore interval method or the improved interval method [5J, it is necessary to use a significantly smaller integration step h < 10" /(D, which leads to the cost of computing time of at least 1 hour.
7ТЛ
The developed methods have been applied in solving various practical problems [8].
In conclusion, ive express our gratitude to doctor of technical Sciences, Professor Nekrasov for assistance in the preparation of the article.
\. Kalmykov S.A., Shokin, Y.I., Yuldasliev Z.H. (1986). Methods of interval analysis, Novosibirsk: Science.
2. Dobronets B.S., Shaidurov V.V. (1990), Bilateral meth-öife.Novosibirsk: Science.
3. Korn G., Korn T. (1978) Handbook of mathematics, Moscow: Science.
4. Nekrasov S.A. (1986). The Bilateral method for solving C'auchy problems. J. comp, mod. and math. Fiz., No. 5.
5. Nekrasov S.A. (1988). On the construction of two-sided approximations to the solution of the Cauchy problem. J. Comput. mod. and math. Phys., No. 5.
6. Nekrasov S.A. (1995). Bilateral methods of integration of initial and boundary value problems. J. Comput. mod. and math. Phys., No. 10.
7. Nekrasov S.A. (2003). Effective two-way methods for solving the Cauchy problem in the case of large intervals of integration. Differential equations, No. 7.
8. Nekrasov S.A, Interval and two-sided methods for calculation with guaranteed accuracy of electric and magnetic systems. Thesis for the degree of doctor of technical Sciences in "Theoretical electrical engineering". Novocherkassk. YURGTU(NPl). 2002.
References
ПРОБЛЕМА НЕУСТОЙЧИВОСТИ ИНТЕРВАЛЬНЫХ МЕТОДОВ ПРИ БОЛЬШИХ ПРОМЕЖУТКАХ ИНТЕГРИРОВАНИЯ И ЕЕ РЕШЕНИЕ
Черноиван Дмитрий Николаевич, Южно-Российский государственный политехнический университет (НПИ),
г. Новочеркасск, Россия, [email protected] Середина Полина Борисовна, Южно-Российский государственный политехнический университет (НПИ),
г. Новочеркасск, Россия, [email protected]
Аннотация
В статье описано решение проблемы принципиальной неустойчивости известных интервальных и двусторонних методов для численного решения ОДУ на больших промежутках интегрирования. Проблема связана с экспоненциальным ростом погрешности и весьма жестким ограничением на длину интервала интегрирования. Рассмотрены примеры иллюстрирующие неустойчивость известных двусторонних методов (классического интервального метода Мура первого порядка, двустороннего метода Б.С.Добронца сколь угодно высокого порядка точности) на больших интервалах интегрирования. Рассмотрены двусторонние методы решения задачи Коши, эффективные в случае больших промежутков интегрирования. По характеристикам сходимости и устойчивости для случая нелинейных задач предложенные методы существенно превосходят известные аналоги при практически равном порядке объема вычислительных затрат на шаге интегрирования. Приводятся соответствующие теоретические оценки и примеры, подтверждающие эффективность предложенных двусторонних методов. Разработанные методы подразделяются на следующие группы. Методы первой группы основаны на апостериорной оценке погрешности определенного вещественного конечно-разностного метода. Качественное улучшение сходимости и устойчивости двустороннего метода достигается за счет использования мажорантных оценок нормы сеточных функций Грина, соответствующих схеме используемого вещественного метода. При использовании двусторонних методов второй группы не имеет принципиального значения, каким методом находится приближенное решение, погрешность которого подлежит апостериорной оценке. Для нахождения оценок также используются мажоранты для нормы функции Грина, однако не сеточной, а непрерывной линеаризованной задачи. Чтобы вычислить мажоранты для функций Грина линейной задачи используется соответствующий двусторонний метод, что усложняет алгоритм вычислений. Поэтому использование методов второй группы целесообразно в тех случаях, когда приближенное решение получено неизвестным численным методом или при помощи экспериментальных измерений.
Ключевые слова: двусторонний метод, интервальный метод, большие промежутки, неустойчивость, погрешности, гарантированная точность.
Литература
1. Калмыков С.А., Шокин Ю.И., Юлдашев З.Х. Методы интервального анализа. Новосибирск: Наука, 1986.
2. Добронец Б.С., Шайдуров В.В. Двусторонние методы. Новосибирск: Наука, 1990.
3. Корн Г., Корн Т. Справочник по математике. М.: Наука, 1978.
4. Некрасов С.А. Двусторонний метод решения задач Коши // Ж. вычисл. матем. и матем. физ., №5, 1986.
5. Некрасов С.А. О построении двусторонних приближений к решению задачи Коши // Ж. вычисл. матем. и матем. физ., №5, 1988.
6. Некрасов С.А. Двусторонние методы интегрирования начальных и краевых задач // Ж. вычисл. матем. и матем. физ., №10, 1995.
7. Некрасов С.А. Эффективные двусторонние методы для решения задачи Коши в случае больших промежутков интегрирования // Дифференциальные уравнения, №7, 2003.
8. Некрасов С.А. Интервальные и двусторонние методы для расчета с гарантированной точностью электрических и магнитных систем. Диссертация на соискание ученой степени доктора технических наук по специальности "Теоретическая электротехника". Новочеркасск. ЮРГТУ(НПИ). 2002.
Информация об авторах:
Черноиван Дмитрий Николаевич, Южно-Российский государственный политехнический университет (НПИ), магистр, г. Новочеркасск, Россия Середина Полина Борисовна, Южно-Российский государственный политехнический университет (НПИ), бакалавр, г. Новочеркасск, Россия
