=Paper=
{{Paper
|id=Vol-1638/Paper63
|storemode=property
|title=On interpolation of functions with a boundary layer by cubic splines
|pdfUrl=https://ceur-ws.org/Vol-1638/Paper63.pdf
|volume=Vol-1638
|authors=Igor A. Blatov,Elena V. Kitaeva,Alexander I. Zadorin
}}
==On interpolation of functions with a boundary layer by cubic splines ==
https://ceur-ws.org/Vol-1638/Paper63.pdf
Mathematical Modeling
ON INTERPOLATION of FUNCTIONS with a
BOUNDARY LAYER BY CUBIC SPLINES
I.A. Blatov1, E.V.Kitaeva1, A.I. Zadorin2
1
Volga Region State University of Telecommunications and Informatics, Samara, Russia
2
Sobolev Institute of Mathematics of Siberian Branch of Russian Academy of Sciences,
Novosibirsk, Russia
Abstract. The problem of article is cubic spline-interpolation of functions hav-
ing high gradient regions. It is shown that uniform grids are inefficient to be
used. In case of piecewise-uniform grids, concentrated in the boundary layer,
for cubic spline interpolation are announced asymptotically exact estimates on a
class of functions with an exponential boundary layer. There are obtained re-
sults showing divergent in small parameter estimates and divergence of interpo-
lation processes. The modified cubic spline with uniform in small parameter in-
terpolation error is offered. The results of numerical experiments confirming
theoretical estimates are given.
Keywords: boundary layer, singular perturbation, cubic spline-interpolation,
error estimates.
Citation: Blatov IA, Kitaeva EV, Zadorin AI. About interpolation by cubic
splines of the functions with a boundary layers. CEUR Workshop Proceedings,
2016; 1638: 515-520. DOI: 10.18287/1613-0073-2016-1638-515-520
Introduction
Cubic splines are widely applied to smooth interpolation of functions. Such splines
are investigated in [1], [2] and in many other works. However, according to [3], [4],
application of the polynomial spline-interpolation to functions with large gradients in
a boundary layer leads to essential errors of O(1) type. In [4] there was constructed a
non-polynomial analogue of cubic spline, which is exact for the boundary layer
component. Numerical experiments have shown the advantage in the accuracy of the
constructed spline. However, the boundary layer component isn't always known, thus,
in such case, there is no reasonable alternative for condensing a grid in the boundary
layer. In this work a traditional cubic spline interpolation [2] on the piecewise uni-
form grid condensing in the boundary layer is investigated. There were obtained error
estimates of interpolation which, however, aren't uniform in small parameter . It is
shown that an interpolation error of a boundary layer component might increase with-
out limits while 0 , thus, we need to develop special methods of interpolation
for such functions. Such a method of interpolation is also offered and investigated in
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this work. We shall pass it noting that the divergence of interpolation processes by
cubic and parabolic splines on nonuniform grids was regarded in works [2], [5], [6]
and some other works. However, the examples of divergence provided there had an
artificial character or were implied with the help of Banach-Steinhaus theorem. In the
present article we have shown divergence for functions describing solutions of a vari-
ety of applied problems. These results testify the need of development of universal
high-order methods of smooth spline interpolation of functions on nonuniform grids
and development of projective-grid methods of a high order for singular perturbed
boundary value problems, because there is no need for application of the grid solution
interpolation while using projective-grid methods.
Statement of the problem
Let us introduce the following notations. Let : 0 x0 x1 x N 1 − parti-
tion of a segment [0,1] . Let S (, k ,1) be a space of polynomial splines [2] of de-
gree k and defect 1 on the grid . We means that C, C j are positive constants
independent from and a number of grid nodes. We write f O(g ) if
f C g and f O * ( g ) , if f O(g ) and g O( f ) , C[a, b] is a space of
continuous functions with norm .
C [ a ,b ]
Let a function u(x) have a form
u( x) q( x) ( x), x [0,1] ,
q ( j ) ( x) C1 , ( j ) ( x) C1e x / / j , 0 j 4 . (1)
Let us investigate a problem of cubic spline-interpolation of the function (1).
Main results
First, let us look into the case of a uniform grid. Let N be a natural number, is uni-
form grid with a step of H 1 / N and nodes xn , n 0,1,, N , partitioning of the
interval [0,1] . Let g 3 ( x, u ) S ( ,3,1) be an interpolation cubic spline on the grid
, defined by conditions:
g 3 ( xn , u ) u ( xn ), 0 n N , g 3 ' (0, u ) u ' (0), g 3 ' (1, u ) u ' (1). (2)
Theorem 1. In case of a uniform grid there shall be such a constant C for which the
next estimate is correct:
u( x) g3 ( x, u) C[0,1] C( N ) 4 .
x /
If in (1) ( x ) e , then the following estimate holds:
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u( x) g3 ( x, u) C[0,1] C1 min ( N ) 1 , ( N ) 4 .
Next, according to [7] let us define the grid with nodes xn , n 0,1,, N , and
steps
N 1 N
hn h , n 1, , , hn H , n 1,, N .
N /2 2 N /2 2
In accordance with [7] let us define
1 4
min , ln N . (3)
2
Due to [2] for interpolation cubic spline g3 ( x, u ) S (,3,1) the following error
estimate is valid:
5 ( 4)
g 3 ( x, u ) u ( x ) u max hn4 . (4)
384 C [ 0 ,1] n
Note that g 3 ( x, u ) g 3 ( x, q ) g 3 ( x, ) , and due to (1),(4)
q( x) g 3 ( x, q) C[0,1] C 2 max hn4 C2 N 4 .
n
Henceforth, in order to build a spline interpolation to u(x) with the order of
O ( N 4 ln 4 N ) , it is needed to satisfy the inequality:
( x) g3 ( x, ) C[0,1] C2 N 4 ln 4 N. (5)
In case when in (5) 1 / 2 , an inequality (4) shall be valid in account of Theorem
1 and due to the relation N O ( N / ln N ) . Thus, we shall propose below that
*
1/ 2 . Yet, to keep it short we shall assign g 3 ( x) g 3 ( x, ),
g 3 ( x) S (,3,1) .
Theorem 2. There are such constants C 2 ,C 3 , which satisfy the relation (5) if
N 1 C3 .
Theorem 3. There are such constants C 4 , C 5 , and 0 , independent from , N ,
that if C 4 N
1
, then
N 4 ln 4 N ,0 n N / 2 1
g 3 ( x, ) ( x) C[ x , x ] C5 N 5 ( n N / 2) . (6)
n n 1
e , N / 2 n N 1
Next Theorem shows us that estimates in (6) can not be improved.
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x/
Theorem 4. Let ( x) e .Then there are such constants C 4 , C6 , 1 0 ,
independent from , N that if C 4 N
1
, then
N 5 N
g 3 ( x, ) ( x) C[ x , x C6 e 1 ( n N / 2) , n N 1
n n 1 ]
2 .
Now we shall construct a modified interpolation spline. Let
x N / 2 ( x N / 2 x N / 2 1 ) / 2, xn xn , n [0, N / 2 1] [ N / 2 1, N ] . Let
gm3 ( x, u ) be the interpolation cubic spline defined by conditions
gm3 ( xn , u ) u ( xn ) , n[0, N ] , gm3 ' (0, u ) u ' (0), gm3 ' (1, u ) u ' (1) .
The only difference between gm3 ( x, u ) and g 3 ( x, u ) is that the interpolation node
x N / 2 is set as x N / 2 . The spline nodes whereas are not subject to any changes and
coincide with the nodes.
Theorem 5. There are such independent from , N constants, namely 0 0, C that
if ln N 0 , it shall satisfy the following inequality :
u( x) gm3 ( x, u) C[0,1] CN 4 ln 4 N. (7)
Comment 1. The condition ln N 0 will be satisfied if CN .
1
Therefore, according to theorems 2,5 application of the interpolation spline
gm3 ( x, u ) if O ( N ) 1
and the interpolation spline g 3 ( x, u ) if
N 1 O( ) let as to obtain estimates (4), (7) uniformly in , N .
Results of Numerical Experiments
Let us define the following function:
x
x
u ( x) cos e , x [0,1]
2 . (8)
Results of calculations are provided in the three following tables. Given in the tables
below are the maximum errors of spline interpolation, calculated at nodes of the con-
densed grid, which is obtained from the initial grid by splitting every single grid in-
terval into 10 parts. Table 1 contains interpolation errors for the traditional cubic
spline on the uniform grid. Results confirm estimates of the Theorem 1 and an inade-
quacy of application of the uniform grid for small values of . Table 2 provides er-
rors of a traditional cubic spline in Shishkin meshes. It follows from the tables that
errors increase while decreases and N is fixed. Results of Table 3 describing the
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errors of modified cubic spline, on the contrary, show uniform convergence, thus,
theoretical conclusions are confirmed.
Table 1. Errors of cubic spline on the uniform grid
ɴ 16 32 64 128 256 512
ɛ
1 2.82e-7 1.76e-8 1.16e-9 1.02e-10 4.30e-11 2.68e-13
10e-1 3.43e-4 2.33e-5 1.51e-6 9.58e-8 6.03e-9 4.11e-10
10e-2 0.43 8.38e-2 9.72e-2 8.00e-4 5.59e-5 3.65e-6
10e-3 9.88 4.58 1.93 0.66 0.15 2.03e-2
10e-4 1.05e+2 5.23e+1 2.58e+1 1.25e+1 5.90 2.59
10e-5 1.06e+3 5.23e+2 2.64e+2 1.32e+2 6.56e+1 3.24e+1
10e-6 1.06e+4 5.30e+3 2.65e+3 1.33e+3 6.62e+2 3.30e+2
10e-7 1.06e+5 5.30e+4 2.65e+4 1.33e+4 6.63e+3 3.30e+3
10e-8 1.06e+6 5.30e+5 2.65e+5 1.33e+5 6.63e+4 3.31e+4
Table 2. Errors of cubic spline on piecewise-uniform grid
ɴ 16 32 64 128 256 512
ɛ
1 2.82e-7 1.76e-8 1.16e-9 1.02e-10 4.30e-11 2.68e-13
10e-1 3.43e-4 2.33e-5 1.51e-6 9.58e-8 6.03e-9 4.11e-10
10e-2 6.43e-3 1.18e-3 1.70e-4 2.07e-5 2.27e-6 2.31e-7
10e-3 6.43e-3 1.18e-3 1.70e-4 2.07e-5 2.27e-6 2.31e-7
10e-4 6.43e-3 1.18e-3 1.70e-4 2.07e-5 2.27e-6 2.31e-7
10e-5 4.47e-2 1.25e-3 1.70e-4 2.07e-5 2.27e-6 2.31e-7
10e-6 4.47e-1 1.25e-2 3.62e-4 2.07e-5 2.27e-6 2.31e-7
10e-7 4.47 1.25e-1 3.62e-3 1.07e-4 3.24e-6 2.31e-7
10e-8 4.47 1.25 3.62e-2 1.07e-3 3.24e-5 9.92e-7
Table 3. Errors of modified cubic spline on piecewise-uniform grid
ɴ 16 32 64 128 256 512
ɛ
1 3.1e-7 2.0e-8 1.3e-9 1.1e-10 4.9e-12 3.1e-13
10e-1 3.4e-4 2.3e-5 1.5e-6 9.6e-8 6.03e-9 4.1e-10
10e-2 6.43e-3 1.2e-3 1.7e-4 2.1e-5 2.3e-6 2.3e-7
10e-3 6.43e-3 1.2e-3 1.70e-4 2.07e-5 2.3e-6 2.3e-7
10e-4 6.43e-3 1.2e-3 1.70e-4 2.07e-5 2.3e-6 2.3e-7
10e-5 6.43e-3 1.2e-3 1.70e-4 2.07e-5 2.3e-6 2.3e-7
10e-6 6.43e-3 1.2e-3 1.70e-4 2.07e-5 2.3e-6 2.3e-7
10e-7 6.43e-3 1.2e-3 1.70e-4 2.07e-5 2.3e-6 2.3e-7
10e-8 6.43e-3 1.2e-3 1.70e-4 2.07e-5 2.3e-6 2.3e-7
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Fig. 1. Graphs of function (8) and of its interpolation spline on the piecewise uniform grid at
the [0,1] interval.
Acknowledgement
The work is supported by Russian Foundation of Basic Researches under Grant 15-
01-06584.
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