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Вычислительные технологии, 2020, том 25, №1, с. 49–65. ©ИВТ СО РАН, 2020 ISSN 1560-7534
Computational Technologies, 2020, vol. 25, no. 1, pp. 49–65. ©ICT SB RAS, 2020 eISSN 2313-691X
COMPUTATIONAL TECHNOLOGIES
DOI:10.25743/ICT.2020.25.1.004
Numerical analysis of grid-clustering rules for problems
with power of the first type boundary layers
V. D. Liseikin1,2, S. Karasulji´
c3
1Institute of Computational Technologies SB RAS, 630090, Novosibirsk, Russia
2Novosibirsk State University, 630090, Novosibirsk, Russia
3University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina
Corresponding author: Liseikin Vladimir D., e-mail: liseikin.v@gmail.com
Received June 10, 2019, revised December 2, 2019, accepted December 18, 2019
This paper demonstrates results of numerical experiments on some popular and new
layer-resolving grids applied for solving one-dimensional singularly-perturbed problems
having power of the first type boundary layers.
Keywords: singularly perturbed equations, small parameter, boundary and interior
layers, grid generation.
Citation: Liseikin V.D., Karasulji´c S. Numerical analysis of grid-clustering rules for
problems with power of the first type boundary layers. Computational Technologies.
2020; 25(1):49–65.
Introduction
The present paper describes experiments on some popular and other forms of layer-resolving
grids — above and beyond those already well known and having broad acceptance, namely,
those developed by Bakhvalov [1], Vulanovi´c [2], and Shishkin [3]. Their grids have been
applied to diverse problems, but only to problems with exponential-type layers [3 – 5], typ-
ically represented by functions exp(−𝑏𝑥/𝜀𝑘) occurring in problems for which the solutions
of reduced (𝜀= 0) problems do not have singularities. Hereinafter 𝑘is the scale of a layer.
The grids of Bakhvalov and Shishkin require knowledge of the constant 𝑏affecting the width
of the exponential layer — when such knowledge is not always available, for example, for
boundary layers in fluid-dynamics problems modelled by Navier— Stokes equations, or for
interior layers in solutions to quasilinear nonautonomous problems. One spectacular exam-
ple of the new layer-resolving grids being presented in the current paper, engendered by a
function 𝜀𝑟𝑘/(𝜀𝑘+𝑥)𝑟,𝑟 > 0, is suitable for dealing not only with exponential layers having
arbitrary widths, but with power of the first type layers occurring in problems for which
the solutions of reduced problems have singularities as well. Another example of a new
layer-resolving grid is aimed at dealing with logarithmic layers represented by a function
ln(𝜀𝑘+𝑥)/ln 𝜀𝑘. It seems that the new layer-resolving grids described in this paper should
empower and spark researchers to solve broader and more important classes of problems hav-
ing not only exponential-, but power-, logarithmic-, and mixed-type boundary and interior
layers.
By the application of algebraic methods or inverted Beltrami and diffusion equations
in control metrics, the layer-resolving grids can be used for solving multidimensional prob-
lems [6].
49
50 V. D. Liseikin, S. Karasulji´c
1. Explicit generation of layer-damping transformations
This section gives a detailed description of basic layer-damping functions near the boundary
point 𝑥0= 0 which are applied to specify global layer-damping transformations and corre-
sponding global layer-resolving grids on the entire interval of calculations with arbitrarily
allocated layers, by the procedures of shifting, blending, scaling, inverting, composing, and
matching them with themselves and polynomial mappings.
1.1. Basic layer-damping transformations
Local contraction transformations 𝑥𝐵(𝜉, 𝜀, 𝑏, 𝑘), 𝑥𝐿𝑖(𝜉, 𝜀, 𝑏, 𝑘), 𝑖 = 2,3,4,have the following
form:
𝑥𝐵(𝜉, 𝜀, 𝑏, 𝑘) = −𝜀𝑘
𝑏ln(1 −𝑑𝜉), 𝑘 > 0, 𝑏 > 0,(1)
𝑥𝐿2(𝜉, 𝜀, 𝑏, 𝑘) = 𝜀𝑘(1 −𝑑𝜉)−1/𝑏 −1, 𝑘 > 0, 𝑏 > 0,(2)
𝑥𝐿3(𝜉, 𝜀, 𝑏, 𝑘) = (𝜀𝑘𝑏 +𝑑𝜉)1/𝑏 −𝜀𝑘, 𝑘 > 0,1>𝑏>0,(3)
𝑥𝐿4(𝜉, 𝜀, 𝑏, 𝑘) = 𝜀𝑘((1 + 𝜀−𝑘)𝑏𝜉 −1), 𝑘 > 0, 𝑏 > 0,(4)
where 𝜀∈(0,1] is a small parameter. Differential equations with the small parameter 𝜀
multiplying the highest-order derivative terms model viscous flows, where 𝜀is typically the
reciprocal of the nondimensional Reynolds number Re; these equations describe problems
of elasticity, where the parameter represents the shell thickness, or simulate flows of liquid
in regions having orifices with a small diameter. As a rule, the solutions of these problems
have highly localized regions (boundary and interior layers) of rapid variation.
The transformation 𝑥𝐵(𝜉, 𝜀, 𝑏, 𝑘), for 𝑘= 1, was introduced by Bakhvalov [1], while the
transformations 𝑥𝐿𝑖(𝜉, 𝜀, 𝑏, 𝑘), 𝑖 = 2,3,4, were introduced by Liseikin [7–9]. A particular
shape of the contraction mapping 𝑥𝐿2(𝜉, 𝜀, 𝑎, 𝑘) for 𝑎= 1, 𝑘 = 1/2, having the form
𝑥𝐿2(𝜉, 𝜀, 1,1/2) = 𝜀1/2𝑑𝜉
1−𝑑𝜉 ,
was proposed by Vulanovi´c [2] to generate grid nodes within some exponential layers of scale
𝑘= 1/2.
The points 𝜉𝑝
𝑖,𝑖= 1,2,3,4,such that the 𝑝th derivative of the mapping 𝑥𝐵(𝜉, 𝜀, 𝑏, 𝑘),
𝑥𝐿𝑖(𝜉, 𝜀, 𝑏, 𝑘) on the corresponding interval [0, 𝜉𝑝
𝑖] is 𝜀-uniformly bounded, and the points
𝑥𝐵(𝜉𝑝
𝑖, 𝜀, 𝑏, 𝑘), 𝑥𝐿𝑖(𝜉𝑝
𝑖, 𝜀, 𝑏, 𝑘), 𝑖 = 2,3,4,which are the widths of the corresponding boundary
layers, are described by the following equations:
𝜉𝑝
1=1−𝜀𝑘/𝑝
𝑑, 𝑥𝐵(𝜉𝑝
1, 𝜀, 𝑏, 𝑘) = 𝜀𝑘𝑝
𝑏ln 𝜀−𝑘,
𝜉𝑝
2=1−𝜀𝑘𝛽
𝑑, 𝛽 =𝑏
1 + 𝑝𝑏, 𝑥𝐿2(𝜉𝑝
2, 𝜀, 𝑏, 𝑘) = 𝜀𝑘(1−𝛽/𝑏)−𝜀𝑘,
𝜉𝑝
3=𝑚, 𝑥𝐿3(𝜉𝑝
3, 𝜀, 𝑏, 𝑘) = (𝜀𝑘𝑏 +𝑑𝑚)1/𝑏 −𝜀𝑘,
𝜉𝑝
4=ln 𝜀−𝑘−𝑝ln[ln(1 + 𝜀−𝑘)]
𝑏ln(1 + 𝜀−𝑘), 𝑥𝐿4(𝜉𝑝
4, 𝜀, 𝑏, 𝑘) = 1
ln1/𝑝(1 + 𝜀−𝑘)−𝜀𝑘.
(5)
Numerical analysis of grid-clustering rules for problems ... 51
Hence, for sufficiently small 𝜀, the widths of these boundary layers are connected by the
following inequalities:
𝑥𝐵(𝜉𝑝
1, 𝜀, 𝑏, 𝑘)≪𝑥𝐿2(𝜉𝑝
2, 𝜀, 𝑏, 𝑘)≪𝑥𝐿4(𝜉𝑝
4, 𝜀, 𝑏, 𝑘)≪𝑥𝐿3(𝜉𝑝
3, 𝜀, 𝑏, 𝑘).
In order to define a boundary-layer damping transformation 𝑥(𝜉, 𝜀, 𝑏, 𝑘) for the target
interval [0, 𝑚] through the use of the local univariate mappings 𝑥𝐵(𝜉, 𝜀, 𝑏, 𝑘), 𝑥𝐿𝑖 (𝜉, 𝜀, 𝑏, 𝑘),
𝑖= 2,3,4, from (1)–(4), specified on the corresponding intervals [0, 𝜉𝑝
𝑖] which will provide
adequate clustering of grid nodes near the boundary point 𝑥0= 0, these mappings need to
be extended continuously or smoothly over the interval [0, 𝑚1] to map it monotonically onto
the interval [0, 𝑚]. This can be done by “gluing” these local nonuniform transformations
𝑥𝐵(𝜉, 𝜀, 𝑏, 𝑘), 𝑥𝐿𝑖(𝜉, 𝜀, 𝑏, 𝑘) to other mappings which are more uniform, for example, poly-
nomial functions. The glued transformation extending 𝑥𝐵(𝜉, 𝜀, 𝑏, 𝑘), 𝑥𝐿𝑖(𝜉, 𝜀, 𝑏, 𝑘) should be
smooth, or at least continuous.
1.2. Local transformations eliminating singularities of high order
This section describes local coordinate transformations 𝑥(𝜉, 𝜀) which eliminate singularities
of arbitrary order in the boundary layer near the point 𝑥0= 0 by specifying coefficients in the
local functions (1)–(4). With the help of high-order approximations, such transformations are
suitable for generating layer-resolving grids 𝑥𝑖=𝑥(𝑖/𝑁, 𝜉), 𝑖= 0,1, 𝑁, providing high-order
𝜀-uniform convergence and interpolations for numerical solutions of singularly-perturbed
equations.
1.2.1. Transformations for exponential singularities
Power transformations. For a function 𝑢(𝑥, 𝜀) whose derivatives up to 𝑛in the vicinity
of the boundary point 𝑥0= 0 (0 ≤𝑥≤𝑚) are estimated by an exponential function and 𝑀,
i. e.,
|𝑢(𝑝)(𝑥, 𝜀)|≤ 𝑀[𝜀−𝑘𝑝 exp(−𝑏𝑥/𝜀𝑘) + 1], 𝑏 > 0,1≤𝑝≤𝑛, 0≤𝑥≤𝑚, (6)
we have that
|𝑢(𝑝)(𝑥, 𝜀)|≤ 𝑀, 1≤𝑝≤𝑛, 𝑚 ≥𝑥≥𝑥𝑛
1=𝑘𝑛𝜀𝑘
𝑏ln(𝜀−1),(7)
while inside the interval [0, 𝑥𝑛
1] the derivatives are not 𝜀-uniformly bounded.
In order to eliminate the exponential singularity (6) using a coordinate transformation,
we rely on the basic contraction function (2) for the construction of nonuniformly clustering
grids within both exponential and power (of the first type) boundary layers. This coordinate
transformation, designated also as 𝑥𝐿2(𝜉, 𝜀, 𝑎, 𝑘)∈𝐶𝑙[0, 𝑚1], 𝑛≥𝑙≥0, has the following
form:
𝑥𝐿2(𝜉, 𝜀, 𝑎, 𝑘) =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
𝑐𝜀𝑘((1 −𝑑𝜉)−1/𝑎 −1),0≤𝜉≤𝜉𝑛
2,
𝑐𝜀𝑘(1−𝛽/𝑎)−𝜀𝑘+𝜀𝑘
(1 −𝑑𝜉)1/𝑎 ′
(𝜉𝑛
2)(𝜉−𝜉𝑛
2)+
+1
2𝜀𝑘
(1 −𝑑𝜉)1/𝑎 ′′
(𝜉𝑛
2)(𝜉−𝜉𝑛
2)2+. . . +
+1
𝑙!𝜀𝑘
(1 −𝑑𝜉)1/𝑎 (𝑙)
(𝜉𝑛
2)(𝜉−𝜉𝑛
2)𝑙+𝑐0(𝜉−𝜉𝑛
2)𝑙+1, 𝜉𝑛
2≤𝜉≤𝑚1,
(8)
52 V. D. Liseikin, S. Karasulji´c
where 𝑑= (1 −𝜀𝑘𝛽)/𝜉𝑛
2;𝑚0> 𝜉𝑛
2>0 (for example 𝜉𝑛
2=𝑚1/2); 0 < 𝑚2≤𝛽=𝑎/(1 + 𝑛𝑎);
𝑎is a positive constant; 𝑛=𝑡+ 1, where 𝑡is the order of the numerical scheme under
consideration; 𝑐 > 0 is such as satisfies a necessary boundary condition 𝑥2(𝑚1, 𝜀, 𝑎, 𝑘) = 𝑚;
and
𝜀𝑘
(1 −𝑑𝜉)1/𝑎 (𝑖)
(𝜉𝑛
2) = 𝑑𝑖1
𝑎1
𝑎+ 1. . . 1
𝑎+𝑖−1𝜀𝑘𝑎(𝑛−𝑖)/(1+𝑛𝑎), 𝑛 ≥𝑖≥1.
In the present paper we use the value 𝑙= 2,for the numerical solving boundary-values
problems having the boundary layer near 𝑥= 0 we use 𝑚1=𝑚= 1,while in the case of two
boundary layers i. e. 𝑥= 0, 𝑥 = 1 we use 𝑚1=𝑚= 1/2,to determinate the constant 𝑐. It
was proved in [10] that the transformation (8) eliminates singularity (6) up to order 𝑛, i. e.,
⃒⃒⃒⃒
d𝑛
d𝜉𝑛𝑢[𝑥𝐿2(𝜉, 𝜀, 𝑎, 𝑘), 𝜀]⃒⃒⃒⃒
≤𝑀, 0≤𝜉≤𝑚1.(9)
Logarithmic transformations. In the same manner it was proved in [10] that, in order
to eliminate locally (in the vicinity of the boundary layer near 𝑥0= 0) the exponential
singularity (6) of the function 𝑢(𝑥, 𝜀) up to order 𝑛in a new coordinate 𝜉, we can use the
basic logarithmic contraction function 𝑥𝐵(𝜉, 𝜀, 𝑎, 𝑘) in the form (1) on the corresponding
interval [0, 𝜉𝑛
1] (see (5)):
𝑥𝐵(𝜉, 𝜀, 𝑎, 𝑘) = −𝜀𝑘
𝑎ln(1 −𝑑𝜉),0≤𝜉≤𝜉𝑛
1,(10)
where 𝑑= (1 −𝜀𝑘/𝑛)/𝜉𝑛
1,but with the restriction 𝑏/𝑛2≥𝑎 > 0, and then prolongate it
smoothly on the interval [0, 𝑚1]. The local transformation of this kind with 𝑘= 1 was
introduced by Bakhvalov [1].
The transformation (8) is more convenient for eliminating exponential singularities than
the transformation (10), since the constant 𝑎in (8) is not dependent on 𝑏in (6), so that,
with an arbitrary fixed constant 𝑎 > 0, this transformation alone is valid for all constants
𝑏∈(0,∞) in (6) for eliminating singularities of 𝑢(𝑥, 𝜀) up to order 𝑛. Another common
piecewise uniform transformation
𝑥𝑆ℎ(𝜉, 𝜀, 𝑏) = ⎧
⎨
⎩
2𝜎𝜉, 0≤𝜉≤1/2,
𝜎+ 2(1 −𝜎)𝜉, 1/2≤𝜉≤1,
(11)
where 𝜎= min{0.5,(𝑛/𝑏)𝜀ln 𝑁}, proposed by Shishkin [3] for generating grids in exponential
layers, is also dependent on constant 𝑏in (6), so that such a grid with a fixed constant will
not be suitable for all 𝑏∈(0,∞) in (6). Compared with the grid of Bakhvalov, the grid of
Shishkin provides less uniform accuracy.
1.2.2. Transformations for power singularities
Transformations for power singularities of the first type. The local power trans-
formation (8) with a proper choice of constant 𝑎 > 0 is also suitable for eliminating power
singularities of the first type near 𝑥0= 0, i. e., when solution derivatives are estimated by
the following formula:
|𝑢(𝑝)(𝑥, 𝜀)| ≤ 𝑀[𝜀𝑘𝑏/(𝜀𝑘+𝑥)𝑏+𝑝+ 1],1≤𝑝≤𝑛, 0≤𝑥≤𝑚. (12)
Numerical analysis of grid-clustering rules for problems ... 53
Here, the boundary-layer interval, where all the derivatives up to 𝑛of 𝑢(𝑥, 𝜀) are not uni-
formly bounded over 𝜀, is [0, 𝑥𝑛
2], 𝑥𝑛
2=𝑚2𝜀𝑘𝑏/(𝑏+𝑛)≫𝑥𝑛
1= (𝑘𝑛/𝑏)𝜀𝑘ln(𝜀−1) for sufficiently
small 𝜀, so that the transformations (10) and (11) may not be suitable for generating layer-
resolving grids for such singularities having incomparably wider layers than any exponential
layer.
It can be proved as in [10] that the transformation (8), but with the following restrictions
𝛽=𝑎/(1 + 𝑛𝑎) and 0 < 𝑎 ≤𝑏/𝑛2, eliminates singularity (12) up to order 𝑛.
Transformations for logarithmic singularities. Solution derivatives near 𝑥0= 0 can
also be estimated by
|𝑢(𝑝)(𝑥, 𝜀)| ≤ 𝑀[1 + 1/((𝜀𝑘+𝑥)𝑝|ln 𝜀|)],1≤𝑝≤𝑛, 0≤𝑥≤𝑚. (13)
Unfortunately, the transformation which would eliminate this singularity up to order 𝑛 >
1 has not yet been found. The following transformation, based on (4), eliminates this
singularity up to order 1 only:
𝑥(𝜉, 𝜀𝑘) =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
𝑐𝜀𝑘1 + 1
𝜀𝑘ln(𝜀−𝑘)𝜉/𝜉0
−1,0≤𝜉≤𝜉0,
𝑐ln−1(𝜀−𝑘) + 2(𝜀𝑘+ ln−1(𝜀−𝑘))×
×ln 1 + 1
𝜀𝑘ln(𝜀−𝑘)(𝜉−𝜉0) + 𝑐0(𝜉−𝜉0)2, 𝜉0≤𝜉≤𝑚1.
(14)
2. Semilinear boundary-value problem
In this section we consider a semilinear boundary-value problem
−(𝜀+𝑟𝑥)𝑢′′ +𝑎(𝑥)𝑢′+𝑓(𝑥, 𝑢)=0,0<𝑥<1, 𝑢(0) = 𝑢0, 𝑢(1) = 𝑢1,(15)
with the following conditions:
0< 𝜀 ≪1, 𝑟 = 0,or 𝑟= 1, 𝑎(𝑥)∈𝐶𝑛[0,1],
𝑓(𝑥, 𝑢)∈𝐶𝑛,𝑛+1([0,1] ×𝑅), 𝑓𝑢(𝑥, 𝑢)≥𝑐 > 0 (16)
for (𝑥, 𝑢)∈[0,1] ×𝑅. This problem, for 𝑟= 0, modells qualitative behavior of viscous flows.
For 𝑟= 1 it was considered in [11]. Solutions to this problem with small 𝜀may have boundary
and interior layers of exponential and power types: they have power boundary layers of the
first type near 𝑥= 0 when 𝑟= 0, 𝑎(0) = 0, 𝑎′(0) >0; or when 𝑟= 1, 𝑎(0) <−1 [12].
Various rules for grid clustering are analyzed in this section for these very cases of power-
type layers. The case of power boundary layers of the second type near interior point 𝑥=𝑥0
was considered in [13] and [10]. Some numerical experiments with schemes of high order
for solving (15) for 𝑟= 0 with various types of layers on the grids defined through (8) were
carried out in [14].
54 V. D. Liseikin, S. Karasulji´c
2.1. Numerical algorithm
We use as an approximation of the singularly-perturbed problem (15) the standard upwind
scheme on a nonuniform grid 𝑥𝑖, 𝑖 = 0,1, . . . , 𝑁,𝑥0= 0 < 𝑥1< . . . < 𝑥𝑁= 1:
−2(𝜀+𝑟𝑥𝑖)
ℎ𝑖+ℎ𝑖−1𝑢ℎ
𝑖+1 −𝑢ℎ
𝑖
ℎ𝑖
−𝑢ℎ
𝑖−𝑢ℎ
𝑖−1
ℎ𝑖−1+𝑎−(𝑥𝑖)𝑢ℎ
𝑖+1 −𝑢ℎ
𝑖
ℎ𝑖
+𝑎+(𝑥𝑖)𝑢ℎ
𝑖−𝑢ℎ
𝑖−1
ℎ𝑖−1
+𝑓(𝑥𝑖, 𝑢ℎ
𝑖)=0,
𝑖= 1,2, . . . , 𝑁 −1, 𝑢ℎ
0=𝑢0, 𝑢ℎ
𝑁=𝑢1,
where ℎ𝑖=𝑥𝑖+1 −𝑥𝑖, and 𝑎±= (𝑎± |𝑎|)/2.The nodes 𝑥𝑖,𝑖= 0, . . . , 𝑁, of the layer-resolving
grid are obtained either explicitly by means of a transformation based on the layer-damping
mappings 𝑥𝐵(𝜉, 𝜀, 𝑎, 𝑘), 𝑥𝐿𝑗 (𝜉, 𝜀, 𝑎, 𝑘), 𝑗= 2,3,4,described in Sect.2, namely,
𝑥𝑖=𝑥𝐵(𝑖ℎ, 𝜀, 𝑎, 𝑘), 𝑥𝑖=𝑥𝐿𝑗 (𝑖ℎ, 𝜀, 𝑎, 𝑘), 𝑖 = 0,1, . . . , 𝑁, ℎ = 1/𝑁.
Calculations of problem (15) are conducted for various values of 𝜀: the results in the
1st — 4nd examples were carried out for the values 10−6,10−14,10−20; in the 5th example
we used the values 10−2,10−8,10−10; while for plotting the graphics we used the values
10−2,10−4,10−6,10−8.For each of these values there are used sequences of grids with dou-
bled numbers of grid steps: 𝑁𝑡= 2𝑡𝑁ℎ,𝑡= 0,1, . . ., where 𝑁ℎis the number for the rough
grid. Usually 𝑁ℎ= 50, 𝑡max = 5,i. e. the calculations are carried out on sequences of five
grids with 𝑁0= 50, 𝑁1= 100, 𝑁2= 200, 𝑁3= 400, 𝑁5= 800. The numerical solution at
the 𝑖th node of the grid related to 𝑁𝑡, is designated by 𝑢𝑁𝑡
𝑖,𝑖= 0,1, . . . , 𝑁𝑡.
For estimating the accuracy of the numerical algorithm, the following characteristics are
introduced:
𝑟𝑡,𝜀 = max
0≤𝑖≤𝑁𝑡
|𝑢𝑁𝑡
𝑖−𝑢𝑁𝑡+1
2𝑖|, 𝑡 = 0,1,..., (17)
and, in the case when the accurate solution 𝑢(𝑥, 𝜀) is known,
∆𝑢𝑡,𝜀 = max
0≤𝑖≤𝑁𝑡
|𝑢(𝑥𝑖, 𝜀)−𝑢𝑁𝑡
𝑖|, 𝑡 = 0,1, . . . (18)
Besides this, one more characteristic is introduced
𝑑𝑢𝑡,𝜀 = max
0≤𝑖≤𝑁𝑡−1|𝑢𝑁𝑡
𝑖+1 −𝑢𝑁𝑡
𝑖|,(19)
which is related to the jump of the numerical solution in the neighboring nodes. The charac-
teristics 𝑟𝑡,𝜀, ∆𝑢𝑡,𝜀 are applied to estimate the order of the accuracy of the numerical solution:
𝛽𝑡
1= log2(𝑟𝑡,𝜀/𝑟𝑡+1,𝜀 ), 𝛽𝑡
2= log2(∆𝑢𝑡,𝜀/∆𝑢𝑡+1,𝜀 ), 𝑡 = 0,1,..., (20)
and, consequently, 𝑑𝑢𝑡,𝜀 to estimate the order of the numerical solution jump in the neigh-
boring nodes
𝛽𝑡
3= log2(𝑑𝑢𝑡,𝜀/𝑑𝑢𝑡+1,𝜀 ), 𝑡 = 0,1, . . . (21)
Notice, if a solution to (15) has neither boundary nor interior layers, then for the numer-
ical solution of this problem through the use of a stable scheme of order 𝑝on the uniform
grid 𝑥𝑖=𝑖ℎ the values 𝛽𝑡
1and 𝛽𝑡
2are close to 𝑝, while 𝛽𝑡
3is close to 1. Recall, that we
use the standard upwind finite difference scheme of order 1, and it is better for all three
characteristics (𝛽𝑡
1, 𝛽𝑡
2, 𝛽𝑡
3) to have a value closer to 1. The aim of the present paper is to
find out wether this property is valid for solving problems with power boundary layers of the
first type by using popular grids and the grids defined through transformations (8) and (14).
Numerical analysis of grid-clustering rules for problems ... 55
3. Numerical experiments
In this section we present results obtained by applying the standard upwind finite difference
scheme (17) on nonuniform grids. For Example 3.1–3.3 and 3.4 we assume 𝑟= 0, while for
Example 3.5 we assume 𝑟= 1.
The analytical solutions of the first and second examples have a single power boundary
layer of the first type and of scale 𝑘= 1/2 near the point 𝑥0= 0,while the solutions of the
third and fourth examples have two power boundary layers of the first type and scale 1/2,
near the points 𝑥0= 0 and 𝑥0= 1,finally the solution of the last i. e. the fifth example has
a single power boundary layer of the first type and scale 1 near the point 𝑥0= 0.
The corresponding transformations for the first three grids according (10), (11), and [2],
which will be used in Example 3.1, 3.2 and 3.5, have the forms given below.
Modified Shishkin grid 1 is given by the transformation
𝑥𝑆ℎ1(𝜉, 𝜀𝑘, 𝑏) = 2𝜎𝜉, 06𝜉61/2,
𝜎+ 2𝜎(𝜉−1/2) + 𝜔(𝜉−1/2)3,1/26𝜉61,(22)
where 𝜎= min 0.5,(𝑛/𝑏)𝜀𝑘ln 𝑁,and 𝜔is chosen so that hold 𝑥𝑆ℎ1(1, 𝜀𝑘, 𝑏) = 1.
Bakhvalov grid is given by
𝑥𝐵(𝜉, 𝜀𝑘, 𝑎) = ⎧
⎪
⎨
⎪
⎩
𝜑(𝜉) := −𝜀𝑘
𝑎ln 1−𝜉
𝑞,06𝜉6𝜏,
𝜑(𝜏) + 𝜑′(𝜏)(𝜉−𝜏), 𝜏 < 𝜉 61,
(23)
where 𝑞∈(0,0.5), 𝑏/𝑛2>𝑎 > 0,and the point 𝜏satisfies 𝜑′(𝜏) = 𝜑(𝜏)−1
𝜏−1.
Vulanovi´c grid is given by
𝑥𝑉 𝑢𝑙(𝜉, 𝜀𝑘, 𝑎) = ⎧
⎨
⎩
𝜑(𝜉) := 𝑎𝜀𝑘𝜉
𝑞−𝜉,06𝜉6𝜏,
𝜑(𝜏) + 𝜑′(𝜏)(𝜉−𝜏), 𝜏 6𝜉61,
(24)
where 𝑞∈(0,0.5),and the point 𝜏is calculated from condition 𝑥𝑉 𝑢𝑙(1, 𝜀𝑘, 𝑎) = 1.
Since the solutions of Example 3.3 and 3.4 have two boundary layers near the points
𝑥= 0 and 𝑥= 1,we will use the grids given by the following formulas.
Modified Shishkin grid 2 is given by
𝑥𝑆ℎ2(𝜉, 𝜀𝑘, 𝑏) = ⎧
⎨
⎩
4𝜎𝜉, 06𝜉61/4,
𝜎+ 4𝜎(𝜉−1/4) + 𝜔(𝜉−1/4)3,1/4< 𝜉 61/2,
1−𝑥𝑆ℎ2(1 −𝜉, 𝜀𝑘, 𝑏),1/2< 𝜉 61,
(25)
and now the parameter 𝜎is defined by 𝜎= min{1/4,(𝑛/𝑏))𝜀𝑘ln 𝑁},and 𝜔is chosen from
the condition 𝑥𝑆ℎ2(1/2, 𝜀𝑘, 𝑏)=1/2.
Modified Bakhvalov grid is given by
𝑥𝐵(𝜉, 𝜀𝑘, 𝑎) = ⎧
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎩
𝜑(𝜉) := −𝜀𝑘
𝑎ln 1−𝜉
𝑞,06𝜉6𝜏,
𝜑(𝜏) + 𝜑′(𝜏)(𝜉−𝜏), 𝜏 < 𝜉 61/2,
1−𝑥𝐵(1 −𝜉, 𝜀𝑘, 𝑎),1/2< 𝜉 61,
(26)
56 V. D. Liseikin, S. Karasulji´c
where the parameter 𝜏is calculated from the condition 𝜑′(𝜏) = 𝜑(𝜏)−1/2
𝜏−1/2,and 𝑞= 1/4.
Modified Vulanovi´c grid is given by
𝑥𝑉 𝑢𝑙(𝜉, 𝜀𝑘, 𝑎) = ⎧
⎪
⎪
⎨
⎪
⎪
⎩
𝜑(𝜉) := 𝑎𝜀𝑘𝜉
𝑞−𝜉,06𝜉6𝜏,
𝜑(𝜏) + 𝜑′(𝜏)(𝜉−𝜏), 𝜏 < 𝜉 61/2,
1−𝑥𝑉 𝑢𝑙(1 −𝜉, 𝜀𝑘, 𝑎),1/2< 𝜉 61,
(27)
again, the parameter 𝜏is calculated from the condition 𝜑′(𝜏) = 𝜑(𝜏)−1/2
𝜏−1/2,and 𝑞= 1/4.
For the plotting some graphics in Example 3.3, we will use modified Shihskin grid given by
𝑥𝑆ℎ(𝜉, 𝜀𝑘, 𝑏) = ⎧
⎨
⎩
4𝜎𝜉, 06𝜉61/4,
𝜎+ 4(1/2−𝜎)(𝜉−1/4),1/4< 𝜉 61/2,
1−𝑥𝑆ℎ(1 −𝜉, 𝜀𝑘, 𝑏),1/2< 𝜉 61,
(28)
here the parameter 𝜎is defined by 𝜎= min{1/4,(𝑛/𝑏))𝜀𝑘ln 𝑁}.
Remark 3.1. In the sequel,the grid given by (8) we designate by 1st Liseikin grid, and the
other grid given by (14) we designate by 2nd Liseikin grid.
Modified 1st Liseikin grid is given by
𝑥𝐿2(𝜉, 𝜀, 𝑎, 𝑘) =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
𝑐1𝜀𝑘((1 −𝑑𝜉)−1/𝑎 −1),06𝜉6𝜉0,
𝑐1𝜀𝑘𝑎𝑛/(1+𝑛𝑎)−𝜀𝑘+𝑑1
𝑎𝜀𝑘𝑎(𝑛−1)/(1+𝑛𝑎)(𝜉−𝜉0)+
+1
2𝑑21
𝑎1
𝑎+ 1𝜀𝑘𝑎(𝑛−2)/(1+𝑛𝑎)(𝜉−𝜉0)2+
+𝑐0(𝜉−𝜉0)3, 𝜉06𝜉61/2,
1−𝑥𝐿2(1 −𝜉, 𝜀, 𝑎, 𝑘),1/26𝜉61,
(29)
where 𝑑= (1 −𝜀𝑘𝑎/(1+𝑛𝑎))/𝜉0, 𝜉0= 1/4, 𝑎is a positive constant subject to 𝑎≥𝑚1>0,
𝑐0>0, and
1
𝑐1
= 2 𝜀𝑘𝑎𝑛/(1+𝑛𝑎)−𝜀𝑘+𝑑
4𝑎𝜀𝑘𝑎(𝑛−1)/(1+𝑛𝑎)+𝑑2
2
1
𝑎1
𝑎+ 1𝜀𝑘𝑎(𝑛−2)/(1+𝑛𝑎)(1/4)2+𝑐0(1/4)3.
Modified 2nd Liseikin grid is given by
𝑥(𝜉, 𝜀𝑘) =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
𝑐1𝜀𝑘⎡
⎣1 + 1
𝜀𝑘ln(𝜀−𝑘)𝜉/𝜉0
−1⎤
⎦,06𝜉6𝜉0,
𝑐1ln−1(𝜀−𝑘) + 2(𝜀𝑘+ ln−1(𝜀−𝑘))×
×ln 1 + 1
𝜀𝑘ln(𝜀−𝑘)(𝜉−𝜉0) + 𝑐0(𝜉−𝜉0)2, 𝜉06𝜉61/2,
1−𝑥(1 −𝜉, 𝜀𝑘),1/26𝜉61,
(30)
Numerical analysis of grid-clustering rules for problems ... 57
where 𝜉0= 1/4, 𝑐0= 1,and
1
𝑐1
= 2 ln−1(𝜀−𝑘) + 1
2(𝜀𝑘+ ln−1(𝜀−𝑘)) ln 1 + 1
𝜀𝑘ln(𝜀−𝑘)+𝑐0
42.
The analytical solutions of 2nd, 4th and 5th examples are unknown, so we are going to
use the characteristics 𝛽𝑡
1and 𝛽𝑡
3given by the formulas (20) and (21), in order to investigate
a behavior of the numerical solutions calculated on the different grids, but the analytical
solutions of the 1st and 3rd examples we know and we are going to use the characteristics
𝛽𝑡
2and 𝛽𝑡
3in our analysis.
Example 3.1. Let us consider semilinear boundary-value problem
−𝜀𝑢′′ −8𝑥(𝑥−1/2)3𝑢′+𝑓(𝑥, 𝑢)=0, 𝑢(0) = 0, 𝑢(1) = 2,(31)
where
𝑓(𝑥, 𝑢) = 𝑢−1 + 𝜀1/2
𝜀1/2+𝑥4𝜀3/2
(𝜀1/2+𝑥)2−16𝜀1/2𝑥(𝑥−1/2)3
𝜀1/2+𝑥+ 2𝑥.
The analytical solution is 𝑢(𝑥, 𝜀) = 2 1−𝜀1/2
𝜀1/2+𝑥1−𝜀1/2
𝜀1/2+ 1.Here we have that
𝑓𝑢(𝑥, 𝑢) = 1 >0, 𝑎(0) = 0, 𝑎′(0) = 1 >0, 𝑎(1) = −1<0, 𝑎′(1/2) = 0,the scale of this
problem is 𝑘= 1/2,and the solution of this problem has a single power boundary layer of
the first type.
This problem we tested on modified Shihskin 1, Bakhvalov, Vulanovi´c, 1st and 2nd Li-
seikin, and Shishkin grids. The first five grids were used to calculate the values of 𝛽2and
𝛽3,beside previously mentioned grids also Shishkin grid was used in making some graph-
ics. Other parameters we use: 𝑛= 2, 𝑎 = 1/8, 𝑐0= 1,𝑐1is defined from the condition
𝑥2(1, 𝜀, 𝑎, 1/2) = 1 for 1st Liseikin grid; 𝑐0= 0 and 𝑐is calculated from the condition
𝑥(𝜉, 𝜀) = 1 for 2nd Liseikin grid; 𝑞= 0.45 for Bakhvalov and Vulanovi´c grids. The data
corresponding this example are in Table 1, and the appropriate graphics are given in Fig. 1, 2.
The graphics of two numerical solutions and two analytical solutions are presented in
Fig. 1 (left). The first numerical solution is calculated by using 1st Liseikin grid with
𝑁= 100 and 𝜀= 10−2,while the second one is calculated also by using 1st Liseikin grid
but 𝑁= 100 and 𝜀= 10−6.Every 2nd points of the graphics of the numerical solutions
are shown. It’s a well-known fact that analytical solutions of singularly-perturbed boundary
problems exhibit rapid changes in the boundary and/or interior layer/layers. Usually those
changes are getting faster when the perturbation parameter 𝜀is smaller. From the graphics
(Fig. 1 — left) we can see the creation of such a layer near the end point 𝑥= 0.The
graphic (the red one), corresponding to the perturbation parameter 𝜀= 10−6is narrower
and we see a faster change of both solutions in the layer than in case of the graphic-blue
one, corresponding to the perturbation parameter 𝜀= 10−2.Also on the same figure the
valid points of 1st and 2nd Liseikin grids are presented, calculated by using two values of
the perturbation parameter (10−2and 10−6). It is easy to notice that the grid points are
condensed in the portion corresponding to the layer, and that this condensation is higher in
the case of grids with a smaller perturbation parameter value.
The graphics of the six numerical solutions obtained by using 1st, 2nd Liseikin, Vulanovi´c,
Bakhvalov, Shishkin grids and modified Shishkin grid 1 are presented in Fig. 1 (right). The
58 V. D. Liseikin, S. Karasulji´c
T a b l e 1. The results of Example 3.1
Modified
𝑁Shishkin 1 Bakhvalov Vulanovi´c 1st Liseikin 2nd Liseikin
𝛽2𝛽3𝛽2𝛽3𝛽2𝛽3𝛽2𝛽3𝛽2𝛽3
𝜀= 10−6
50 1.1361 0.6637 0.7468 0.6459 0.8746 0.8701 0.9709 0.9680 1.0193 0.9169
100 1.1454 0.7361 0.5383 0.4209 0.9309 0.9387 0.9849 0.9840 1.0090 0.9602
200 1.0374 0.7863 0.7334 0.4599 0.9642 0.9418 0.9926 0.9917 1.0043 0.9805
400 0.8617 0.8211 0.8317 0.5465 0.9815 0.9697 0.9963 0.9959 1.0022 0.9903
800 0.8688 0.8458 0.9184 0.7495 0.9871 0.9893 0.9981 0.9979 1.0011 0.9952
𝜀= 10−14
50 0.3452 0.5710 0.6457 0.3448 0.0952 0.4975 0.9529 0.9413 1.0593 0.7425
100 0.9459 0.5078 0.3271 0.2533 0.6880 0.6875 0.9729 0.9711 1.0249 0.8884
200 0.6648 0.5392 0.1485 0.2328 0.7873 0.7866 0.9873 0.9847 1.0101 0.9476
400 0.9594 0.9042 0.1158 0.2170 0.8614 0.8607 0.9938 0.9927 1.0059 0.9745
800 1.0561 0.9718 0.1043 0.2054 0.9188 0.8984 0.9968 0.9963 1.0028 0.9874
𝜀= 10−20
50 0.2122 0.5295 0.4402 0.2741 0.0905 0.3967 0.9390 0.9300 1.0975 0.5790
100 0.1837 0.1829 0.2836 0.2345 0.3857 0.5857 0.9711 0.9626 1.0378 0.8264
200 0.1706 0.1625 0.2641 0.2121 0.4831 0.6831 0.9855 0.9826 1.0135 0.9217
400 0.2370 0.1691 0.2483 0.1938 0.5536 0.7535 0.9928 0.9915 1.0040 0.9626
800 0.8333 0.3607 0.2082 0.1791 0.6040 0.8040 0.9964 0.9958 1.0040 0.9816
Fig. 1. The graphics of the analytical and numerical solutions on 1st Liseikin grid 𝑁= 100, 𝜀 =
10−2,10−6(left), the graphics of the numerical solutions on all considered grids but plotted by
using the uniform grid 𝑁= 100, 𝜀 = 10−4(right)
last four grids are constructed to resolve exponential layers, 1st Liseikin grid is constructed
to resolve a power layer of the first type, and 2nd Liseikin grid has a goal to resolve a
logarithm layer. In short the purpose of using layer grids to resolve layers is to reduce
the distance between the grid points in parts of the domain where the analytical solution
changes rapidly. Reasons for reducing the distances have been analyzed many times in the
literature by Bakhvalov, Liseikin, Gartland, Shishkin, and others. In constructing layer-
Numerical analysis of grid-clustering rules for problems ... 59
resolving grids usually we use simple-model problems exhibiting an appropriate layer with
known analytical solutions, and inverse functions corresponding these analytical solutions.
Using these inverse functions we generate the grid points in the layers. Taking into the
account previously written, in the case when the numerical solution was calculated by using
a layer-resolving grid but plotted by using the uniform grid, we expect that the graphic
of this numerical solution corresponding to the layer, be close to the graphic of the linear
function.
The graphics plotted on a described way, 𝑁= 100, 𝜀 = 10−4,and every 2nd points are
presented in Fig. 1 (right). The grid better constructed for a specific problem, gives the
graph that is closer to the graphic of linear function in the part of grid corresponding the
layer. We used in this example 0.5𝑁of the grid points to resolve the layers for 1st, 2nd
Liseikin, Shishkin grids and modified Shishkin grid 1 and slightly less than 0.5𝑁of the grid
points in the same purpose for Bakhvalov and Vulanovi´c grids. From Fig. 1 (right), we see
that graphic closest to the graphic of linear function, is the graphic obtained by using 1st
Liseikin grid. The graphics obtained by using Shiskin, modified Shishkin 1, Bakhvalov and
Vulanovi´c grids suddenly change in the part corresponding the layer, this kind of behavior
we can explain by the fact that these four grids were constructed to resolve a “sharper”
exponential layer. The graphic obtained by using 2nd Liseikin grid also deviates from the
graphic of linear function but this graphic doesn’t have abrupt changes.
Figure 2 (left) shows the graphics of the parts of the numerical solutions and the graphics
of the part of the analytical solutions near the end point 𝑥= 0,of the problem given
in Example 3.1. To calculate the numerical solutions we used 𝑁= 100, 𝜀 = 10−8,and
every 2nd point are plotted. Each graph of the numerical solutions (except the first one) is
moved up 1 cm from the bottom one in order to better transparency. In the same way the
graphic of the analytical solution is shifted and plotted, i. e. the graphics of the numerical
and analytical solutions are plotted together in order to compare. As mentioned a few time
before, Bakhvalov, Shishkin, Vulanovi´c grids and their variations were constructed to resolve
exponential layers, a well-known fact is that a exponential layer comparing to a power layer
of the first type is “sharper”. The grid points intended for a layer are too condensed near
the end point, and they resolve just a part of the layer. The rest of the layer is not divided
by a sufficient number of grid points, this results in higher error values. Grouping of the grid
points near the end point 𝑥= 0 is especially evident in Bakhvalov and Shishkin grids, the
situation with Vulanovi´c and modified Shishkin 1 grids is less worse, while the points of 1st
and 2nd Liseikin grids are much better distributed (Fig. 2 — left). These are the expected
results because Liseikin grids were constructed for boundary-value problems having wider
layers, and don’t have gaps in the distribution of the graphics points, unlike the four grids
mentioned previously.
Figure 2 (right) shows the graphics of the part of the numerical solutions obtained by
using the uniform and 1st Liseikin grids, and the graphic of the analytical solutions. We may
see that the error value is larger in case of using the uniform grid for this semilinear boundary-
value problem. This example demonstrates the justification for using layer-resolving grids
over uniform grids for the numerical solving of problems of these type.
In Table 1 the results of Example 3.1 are presented, i. e. the values of 𝛽2and 𝛽3.
In our experiments we use the standard upwind finite difference scheme. Let us recall
that the order of accuracy of this scheme on a uniform grid for the numerical solving of
boundary-value problems without boundary or interior layers is 1. We expect that 𝛽2and 𝛽3
have a value of approximately 1. We performed calculations of the numerical solutions for
60 V. D. Liseikin, S. Karasulji´c
Fig. 2. The graphics of the parts of the numerical solutions on all considered grids 𝑁= 500, 𝜀 =
10−8(left), the graphic of the parts of the numerical solutions on 1st Liseikin and the uniform grids
𝑁= 200, 𝜀 = 10−8and the graphic of the part of the analytical solution 𝜀= 10−8(right)
𝑁= 50,100,200,400,800,and 𝜀= 10−6,10−14,10−20.From Table 1 we see that only the
1st and the 2nd Liseikin grids give satisfactory values for 𝛽2and 𝛽3for all used values of
𝑁and 𝜀. Vulanovi´c grid gives also the satisfactory values of 𝛽2and 𝛽3for 𝜀= 10−6,but
decreasing the value of the perturbation parameter 𝜀, Vulanovi´c grid becomes useless for
the numerical solving of this boundary-value problem. Also the modified Shishkin 1 and
Bakhvalov grids give unsatisfactory values of 𝛽2and 𝛽3.
Example 3.2. Let us consider the following problem
−𝜀𝑢′′ −8𝑥(𝑥−1/2)3𝑢′+ 1/2𝑢=𝑒𝑥𝑝(𝑥),0<𝑥<1, 𝑢(0) = 0, 𝑢(1) = 1.(32)
We don’t know the analytical solution of this problem, and here we calculated 𝛽1instead 𝛽2.
The results are very similar to the previous one, and here we gave only the results for the
smallest value of 𝜀in Table 2.
Example 3.3. Let us consider the following problem
−𝜀𝑢′′ + 8𝑥(𝑥−1/2)3(𝑥−1) + 𝑓(𝑥, 𝑢) = 0,0< 𝑥 < 1, 𝑢(0) = 2, 𝑢(1) = 0,
T a b l e 2. The results of Example 3.2
Modified
𝑁Shishkin 1 Bakhvalov Vulanovi´c 1st Liseikin 2nd Liseikin
𝛽1𝛽3𝛽1𝛽3𝛽1𝛽3𝛽1𝛽3𝛽1𝛽3
𝜀= 10−20
50 0.3211 0.8908 0.1861 0.9266 0.6048 0.9266 0.8332 0.8508 0.6776 0.8444
100 0.2257 0.9085 0.1733 0.5925 0.5732 0.9526 0.9153 0.9490 0.7684 0.9601
200 0.2807 0.9554 0.1469 0.1577 0.5468 0.9802 0.9573 0.9725 0.9100 0.9534
400 0.5612 0.6085 0.1222 0.1411 0.5288 0.9898 0.9789 0.9856 0.9643 0.9587
800 0.2936 0.4968 0.1020 0.1297 0.5181 0.8107 0.9899 0.9926 0.9845 0.9775
Numerical analysis of grid-clustering rules for problems ... 61
where
𝑓(𝑥, 𝑢)= 𝑢−1−2𝜀3/2(1 + 𝜀1/2)1
(𝜀1/2+𝑥)3−1
(𝜀1/2+ 1 −𝑥)3−8𝜀1/2(1 + 𝜀1/2)𝑥(𝑥−1)×
×(𝑥−1/2)31
(𝜀1/2+𝑥)2+1
(𝜀1/2+ 1 −𝑥)2+𝜀1/2(1 + 𝜀1/2)1
𝜀1/2+𝑥−1
𝜀1/2+ 1 −𝑥.
The analytical solution of this problem is
𝑢(𝑥, 𝜀) = 1 +
𝜀1/2
𝜀1/2+𝑥−𝜀1/2
𝜀1/2+ 1 −𝑥
1−𝜀1/2
𝜀1/2+ 1
.
Here we have 𝑓𝑢(𝑥, 𝑢) = 1 >0, 𝑎(0) = 0, 𝑎′(0) = 1 >0, 𝑎(1) = 0, 𝑎′(1) = 1 >0and
𝑎(1/2) = 0, 𝑎′(1/2) = 0 >0,the scale of this problem is 𝑘= 1/2,and the analytical solution
of this problem has two power boundary layers of the first type near the points 𝑥= 0, 𝑥 = 1.
In our calculation we use: 𝑛= 2, 𝑎 = 1/8and 𝑐0= 1,the parameter 𝑐1is calculated
from the condition 𝑥𝐿2(1/2, 𝜀, 𝑎, 𝑘) = 1/2for 1st Liseikin grid; the parameter 𝑐is calculated
from the condition 𝑥(1/2, 𝜀)=1/2for 2nd Liseikin grid; and 𝑞= 1/4for Bakhvalov and
Vulanovi´c grids. The data corresponding this example is presented in Table 3.
In Figure 3 the parts of the numerical solutions as well as the parts of the analytical
solution are presented. The five graphics of the parts of the numerical and analytical solutions
have been moved up for better transparency. The points of the numerical solutions obtained
by using mod. Shishkin, mod. Shishkin 2, mod. Bakhvalov and mod. Vulanovi´c grids are
highly concentrated near the ends points 𝑥= 0,and 𝑥= 1.We can explain this phenomenon
very easy, these grids are constructed to resolving an exponential layer, not a power layer of
the first type.
Table 3 shows the values of 𝛽2and 𝛽3.Because the analytical solution is known, we
calculated the value of 𝛽2rather than 𝛽1.Also, here the results are better whenever values
Fig. 3. The graphics of the parts of the numerical and analytical solutions 𝑁= 500, 𝜀 = 10−8
62 V. D. Liseikin, S. Karasulji´c
T a b l e 3. The results of Example 3.3
Modified
𝑁Shishkin 1 Bakhvalov Vulanovi´c 1st Liseikin 2nd Liseikin
𝛽2𝛽3𝛽2𝛽3𝛽2𝛽3𝛽2𝛽3𝛽2𝛽3
𝜀= 10−6
50 1.1284 0.5678 0.7682 0.6107 0.5977 0.6766 0.9358 0.9214 1.0575 0.7839
100 0.9543 0.6754 0.4918 0.4375 0.7515 0.7220 0.9642 0.9632 1.0247 0.9013
200 0.8885 0.7501 0.6869 0.4697 0.8940 0.8813 0.9830 0.9818 1.0129 0.9527
400 0.8808 0.8003 0.8288 0.5432 0.9440 0.9178 0.9917 0.9909 1.0060 0.9768
800 0.8798 0.8340 0.8991 0.7198 0.9574 0.9245 0.9959 0.9955 1.0031 0.9885
𝜀= 10−14
50 0.2940 0.5677 0.5529 0.3423 0.4567 0.4111 0.8578 0.8463 1.0591 0.6651
100 0.4823 0.6553 0.3460 0.2761 0.5712 0.5712 0.9352 0.9336 1.0909 0.7098
200 0.8335 0.7501 0.2778 0.2521 0.7027 0.7026 0.9710 0.9660 1.0362 0.8485
400 0.8815 0.8003 0.1290 0.2327 0.7954 0.7953 0.9858 0.9831 1.0211 0.9284
800 0.9722 0.8340 0.1158 0.2179 0.8000 0.7942 0.9923 0.9878 1.0103 0.9651
𝜀= 10−20
50 0.2872 0.5677 0.3711 0.2827 0.4421 0.4001 0.8408 0.8470 1.0128 0.6518
100 0.2013 0.6753 0.3042 0.2575 0.5671 0.5701 0.9418 0.9249 1.1255 0.7082
200 0.1729 0.3751 0.2825 0.2317 0.7002 0.6904 0.9643 0.9588 1.0639 0.7609
400 0.1719 0.1546 0.2638 0.2100 0.7532 0.7894 0.9835 0.9588 1.0289 0.8901
800 0.3322 0.1914 0.2487 0.1924 0.7986 0.7900 0.9908 0.9808 1.0156 0.9472
of 𝛽2and 𝛽3are closer to 1. It’s also desirable that values of 𝛽2and 𝛽3are getting closer
and closer to 1, when the number of points are increasing. Based on the presented results
in Table 3 and just mentioned above, we obtained the best results of 𝛽2and 𝛽3by using 1st
and 2nd Liseikin grids.
Example 3.4. Let us consider the following problem
−𝜀𝑢′′ + 8𝑥(𝑥−1/2)3(𝑥−1)𝑢′+ 0.5𝑢= sin 𝑥, 0<𝑥<1, 𝑢(0) = 1, 𝑢(1) = −1.(33)
We don’t know the analytical solution of this problem, and here we calculated the values of 𝛽1
instead 𝛽2.The results are very similar to the previous one, and we here gave only the results
for the smallest value of 𝜀in Table 4.
T a b l e 4. The results of Example 3.4
Modified
𝑁Shishkin 1 Bakhvalov Vulanovi´c 1st Liseikin 2nd Liseikin
𝛽1𝛽3𝛽1𝛽3𝛽1𝛽3𝛽1𝛽3𝛽1𝛽3
𝜀= 10−20
50 0.3567 0.3241 0.8155 0.3031 0.7846 0.5462 0.8292 0.7621 0.8269 0.6019
100 0.3032 0.5049 0.1689 0.1848 0.5267 0.5188 0.9088 0.8666 1.1309 0.6882
200 0.2477 0.6481 0.1494 0.1637 0.5253 0.5156 0.9496 0.9350 1.2372 0.8643
400 0.3745 0.7426 0.1284 0.1479 0.5101 0.5118 0.9758 0.9652 1.0861 0.9722
800 0.5150 0.4358 0.1097 0.1361 0.5149 0.5092 0.9876 0.9829 1.0253 1.0012
Numerical analysis of grid-clustering rules for problems ... 63
Example 3.5. Let us consider the following problem
−(𝜀+𝑥)𝛼𝑢′′ −(1.2)𝑢′+𝑢=−sin(10𝑥),0<𝑥<1, 𝑢(0) = 0, 𝑢(1) = 1,
where 𝛼= 1.Here is 𝑓𝑢(𝑥, 𝑢)=1>0,−𝑎(0) = 1.2>1,the scale of this problem is 𝑘= 1,
and the analytical solution of this problem has a single power boundary layer power of the first
type near 𝑥= 0.The parameters we used have the values: 𝑛= 2, 𝑎 = 1/20, 𝑐0= 1,and 𝑐1
T a b l e 5. The results of Example 3.5
Modified
𝑁Shishkin 1 Bakhvalov Vulanovi´c 1st Liseikin 2nd Liseikin
𝛽1𝛽3𝛽1𝛽3𝛽1𝛽3𝛽1𝛽3𝛽1𝛽3
𝜀= 10−2
50 0.2134 0.8906 0.8762 1.0182 0.8186 1.0214 0.8190 1.0086 0.8648 1.0077
100 0.1665 0.6049 0.9383 1.0064 0.9094 1.0178 0.9069 1.0049 0.9323 1.0043
200 0.1416 0.6974 0.9696 1.0071 0.9552 1.0079 0.9534 1.0026 0.9664 1.0024
400 0.1303 0.7654 0.9850 1.0022 0.9778 1.0064 0.9767 1.0013 0.9833 1.0011
800 0.1258 0.8128 0.9926 1.0019 0.9889 1.0019 0.9884 1.0006 0.9917 1.0006
𝜀= 10−8
50 -0.035 1.0127 0.0281 0.3130 0.0221 0.3239 0.6543 1.0201 0.6333 1.0228
100 -0.049 1.0079 0.0532 0.2275 0.0347 0.2492 0.6936 1.0167 0.8321 1.0157
200 -0.113 0.9575 0.0577 0.1725 -0.003 0.2166 0.7543 1.0112 0.8898 1.0067
400 -0.053 0.2885 0.0543 0.1352 -0.135 0.2274 1.0465 1.0043 1.0237 1.0019
800 0.3328 0.5200 0.0330 0.1086 -0.356 0.3082 1.0902 1.0014 1.0311 1.0006
𝜀= 10−10
50 -0.028 1.0127 0.0282 0.3130 0.0277 0.3141 0.6148 1.0187 0.6145 1.0218
100 -0.010 1.0079 0.0534 0.2274 0.0523 0.2296 0.6368 0.8017 0.7957 1.0182
200 -0.008 0.9264 0.0581 0.1723 0.0555 0.1766 0.7267 1.2263 0.8268 1.0096
400 -0.030 0.1492 0.0551 0.1348 0.0481 0.1434 1.0414 1.0071 1.0742 1.0025
800 -0.105 0.1141 0.0341 0.1079 0.0253 0.1252 1.2276 1.0020 1.0791 1.0006
Fig. 4. The graphics of numerical solutions 𝑁= 100, 𝜀 = 10−4(left), the graphics of numerical
solutions obtained by using all considered grids, but plotted using the uniform grid (right), for both
figures the parameters have the values 𝑁= 100, 𝜀 = 10−4
64 V. D. Liseikin, S. Karasulji´c
is calculated from the condition 𝑥𝐿2(1/2, 𝜀, 𝑎, 𝑘)=1/2for 1st Liseikin grid; 𝑐is calculated
from the condition 𝑥(1/2, 𝜀)=1/2for 2nd Liseikin grid, and 𝑞= 0.45 for Bakhvalov and
Vulanovi´c grids. The data of this example are in Table 5, and the appropriate graphics are
given in Fig. 4.
Based on the data given in Table 5, we can conclude that the numerical solutions calcu-
lated on 1st and 2nd Liseikin grids are better than other numerical solutions we calculated.
The scale of the layer in this example is 1, and from Fig. 4 (left) we can notice a faster
change of the numerical solutions in the layer, comparing by the previous examples for the
same value of the parameter 𝜀. Bearing in mind our earlier remarks, now from Fig. 4 (right)
we can see that the numerical solutions calculated on 1st and 2nd Liseikin grids are better
than the rest.
Conclusion
In this paper we have calculated the numerical solutions for one-dimensional singularly-
perturbed problems having power boundary layers of the first type by using different layer-
resolving grids. Our goal is to compare the new layer-resolving grids with well-known grids
and show a benefit of using the new layer-resolving grids over to other mentioned grids.
In order to do this, we tested five different examples on six layer-resolving grids. All the
results are presented in the tables. The expected value for all of the examined characteristics
(i. e. 𝛽𝑡
1, 𝛽𝑡
2, 𝛽𝑡
3) is 1. From the tables we can see that the weakest results were demonstrated
by using Shishkin‘s grid (1 or 2), and that the best results were obtained by using the new
layers-resolving grids proposed by Liseikin. These properties of grids are mainly manifested
for the cases of very small values of the parameter 𝜀.
Acknowledgements. The reported study was funded by RFBR, project No. 20-01-00231.
References
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Вычислительные технологии, 2020, том 25, №1, с. 49–65. ©ИВТ СО РАН, 2020 ISSN 1560-7534
Computational Technologies, 2020, vol. 25, no. 1, pp. 49–65. ©ICT SB RAS, 2020 eISSN 2313-691X
ВЫЧИСЛИТЕЛЬНЫЕ ТЕХНОЛОГИИ
DOI:10.25743/ICT.2020.25.1.004
Численный анализ законов сгущения сеток для задач со степенными
погранслоями первого типа
В. Д. Лисейкин1,2, С. Карасулич3
1Институт вычислительных технологий СО РАН, Новосибирск, Россия
2Новосибирский государственный университет, Новосибирск, Россия
3Университет Тузлы, 75000 Тузла, Босния и Герцеговина
Контактный автор: Лисейкин Владимир Д., e-mail: liseikin.v@gmail.com
Поступила 10 июня 2019 г., доработана 2 декабря 2019 г., принята в печать 18 декабря 2019 г.
Аннотация
В статье приведены результаты численных расчетов обыкновенных сингулярно-возмущен-
ных задач, решения которых имеют степенные, первого типа, пограничные слои. Расчеты
проведены с использованием как известных адаптивных сеток, сгущающихся в слоях, так и
новых. Численные эксперименты демонстрируют преимущество новых сеток.
Ключевые слова: уравнение с малым параметром, погранслой, адаптивная сетка.
Цитирование: Лисейкин В.Д., Карасулич С. Численный анализ законов сгущения сеток
для задач со степенными погранслоями первого типа. Вычислительные технологии. 2020;
25(1):49–65. (На англ. яз.)
Благодарности. Исследование выполнено при финансовой поддержке РФФИ в рамках на-
учного проекта №20-01-00231.
