ArticlePDF Available

ON CONSTRUCTION OF A GLOBAL NUMERICAL SOLUTION FOR A SEMILINEAR SINGULARLYPERTURBED REACTION DIFFUSION BOUNDARY VALUE PROBLEM

Authors:
Samir Karasuljic at University of Tuzla
Samir Karasuljic
Hidajeta Ljevaković
Hidajeta Ljevaković
Download full-text PDF
Read full-text
Download citation

Abstract

A class of different schemes for finding the numerical solution of semilinear singularly--perturbed reaction--diffusion boundary--value problems was constructed. The stability of the difference schemes was proved, and the existence and uniqueness of a numerical solution were shown. After that, the uniform convergence with respect to a perturbation parameter ε\varepsilon on a modified Shishkin mesh of order 2 has been proven. For such a discrete solution, a global solution based on a linear spline was constructed, also the error of this solution is in expected boundaries. Numerical experiments at the end of the paper, confirm the theoretical results. The global solutions based on a natural cubic spline, and the experiments with Liseikin, Shishkin and modified Bakhvalov meshes are included in the numerical experiments as well.
Content uploaded by Samir Karasuljic
Author content
All content in this area was uploaded by Samir Karasuljic on Feb 08, 2021
Content may be subject to copyright.
Matematiqki Bilten
ε
ε2y00 −f(x, y) = 0, x ∈(0,1), y(0) = y(1) = 0,
∂f (x, y)
∂y := fy>m > 0,
ε m y
x∈[0,1] f f(x, y)∈
Ck([0,1] ×R), k >2.
x= 0 x= 1.
ε,
ε.
ε
f,
ε
y
s r
y
y=r+s,
j= 0,1, ..., k + 2 x∈[0,1]
r(j)(x)≤C,
s(j)(x)≤Cε−je−x
ε√m+e−1−x
ε√m.
x= 0 x= 1.
0 = x0< x1< . . . < xN= 1,
xi=ψ(i/N).
N+ 1 q∈(0,1/2)
λ:= min 2εln N
√m,1
4.
ψ(t) =
4λt, t ∈[0,1/4],
p(t−1/4)3+ 4λt, t ∈[1/4,1/2],
1−ψ(1 −t), t ∈[1/2,1],
p ψ(1/2) = 1/2, p = 32(1 −4λ). ψ ∈C1[0,1]
kψ0k∞6C, kψ00k∞6C. hi=xi+1−xi, i = 0,...N−1
hi=Z(i+1)N
i/N
ψ0(t) d t6CN −1,
|hi+1 −hi|=Zi/N
(i−1)/N Zt+1/N
t
ψ00(s) d s
6CN −2.
ψ(t) =
4λt, t ∈[0,1/4]
λ+ 2(1 −2λ)(t−1/4), t ∈[1/2,1/4]
1−ψ(1 −t), t ∈[1/2,1].
ψ(t) =
µ(t) := aεt
q−t, t ∈[0, α],
µ(α) + µ0(α)(t−α), t ∈[α, 1/2],
1−ψ(1 −t), t ∈[1/2,1],
a q ε, q ∈(0,1/2), a ∈(0, q/ε),
a√m>2. α
(1/2,1/2) µ(t),
α=q−paqε(1 −2q+ 2aε)
1+2aε .
ψ(t) =
c1εk((1 −dt)−1/a −1),06t61/4,
c1[εkan/(1+na)−εk+d1
aεka(n−1)/(1+na)(t−1/4)
+1
2d21
a(1
a+ 1)εka(n−2)/(1+na)(t−1/4)2
+c0(t−1/4)3],1/46t61/2,
1−ψ(1 −t),1/26t61,
d= (1 −εka/(1+na))/(1/4), a a ≥m1>0
a= 1, c0>0n= 2, k = 1, c0= 0,1
c1
= 2 εkan/(1+na)−εk
+d
4aεka(n−1)/(1+na)+d2
2
1
a(1
a+ 1)εka(n−2)/(1+na)(1/4)2+c0(1/4)3
0 = x0< x1< . . . < xN= 1,
hi=xi+1 −xi, i = 0,1, . . . , N −1.
β
sinh(βhi−1)yi−1−β
tanh(βhi−1)+β
tanh(βhi)yi+β
sinh(βhi)yi+1
=1
ε2"Zxi
xi−1
uII
i−1ψ(s, y) d s+Zxi+1
xi
uI
iψ(s, y) d s#,
i= 1,2, . . . , N −1, y0=yN= 0,
ψ(s, y) = f(s, y)−γ y, β =√γ
ε,
uI
i, uII
i
ε2u00
i−γui= 0 (xi, xi+1),
ui(xi)=1, ui(xi+1 )=0,
i= 0,1, ..., N −1,
ε2u00
i−γui= 0 (xi, xi+1),
ui(xi)=0, ui(xi+1 )=1,
i= 0,1, ..., N −1,
y
ψ ψ
ψ−
i= (1 −t)ψ(xi−1, y(xi−1)) + tψ(xi, y(xi)), x ∈[xi−1, xi],
ψ+
i=tψ(xi, y(xi)) + (1 −t)ψ(xi+1 , y(xi+1)), x ∈[xi, xi+1], t ∈[0,1].
Zxi
xi−1
uII
i−1ds=cosh(βhi−1)−1
βsinh(βhi−1),
Zxi+1
xi
uI
ids=cosh(βhi)−1
βsinh(βhi),
(1 −t) cosh(βhi−1) + t
sinh(βhi−1)(yi−1−yi)−(1 −t) cosh(βhi) + t
sinh(βhi)(yi−yi+1 )
−(1 −t)fi−1+tfi
γ·cosh(βhi−1)−1
sinh(βhi−1)−tfi+ (1 −t)fi+1
γ·cosh(βhi)−1
sinh(βhi)= 0,
i= 1,2, . . . , N −1fk=f(xk, y k), k ∈ {i−1, i, i + 1}, t ∈[0,1].
T,
T u = (T u0, T u1, . . . , T uN)T,
T u0=−u0,
T ui=γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(βhi)−1
sinh(βhi)
·(1 −t) cosh(βhi−1) + t
sinh(βhi−1)(ui−1−ui)−(1 −t) cosh(βhi) + t
sinh(βhi)(ui−ui+1 )
−(1 −t)fi−1+tfi
γ·cosh(βhi−1)−1
sinh(βhi−1)−tfi+ (1 −t)fi+1
γ·cosh(βhi)−1
sinh(βhi)
= 0,
i= 1, . . . , N −1,
T uN=−uN, fk=f(xk, uk), k ∈ {i−1, i, i + 1}.
T y = 0,
y= (y0, y1, . . . , yN)T
y γ >fy.
v, w ∈RN+1
kv−wk6CkT v −T wk.
T y = 0 k(T0y)−1k6C, T 0
T. H := T0(y)
H= [hij ].
h1,1=hN+1,N+1 =−1<0,
hi,i−1=Λ·"(1 −t) cosh(βhi−1) + t
sinh(βhi−1)−
∂f
∂yi−1
γ·(1 −t)(cosh(βhi−1)−1)
sinh(βhi−1)#,
hi,i+1 =Λ·"(1 −t) cosh(βhi) + t
sinh(βhi)−
∂f
∂yi+1
γ·(1 −t)(cosh(βhi)−1)
sinh(βhi)#,
hi,i =−Λ·(1 −t) cosh(βhi−1) + t
sinh(βhi−1)+(1 −t) cosh(βhi) + t
sinh(βhi)
+t
∂f
∂yi
γ·cosh(βhi−1)−1
sinh(βhi−1)+t
∂f
∂yi
γ·cosh(βhi)−1
sinh(βhi)#, i = 2, . . . , N,
Λ=γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(βhi)−1
sinh(βhi)
.
hi,i−1>0, hi,i+1 >0, hi,i <0,
|hi,i|−|hi,i−1|−|hi,i+1 |>m,
H M
kH−1k61
m.
T
RN+1 0T,
T v −T w = (T0ξ)−1(v−w)
ξ= (ξ0, ξ1, . . . , ξN)T∈RN+1. v −w= (T0ξ)−1(T v −T w )
kv−wk=k(T0ξ)−1(T v −T w)k61
mkT v −T wk.
(1 −t)cosh(βhi−1)−1
sinh(βhi−1)(yi−1−yi)−cosh(βhi)−1
sinh(βhi)(yi−yi+1 )
+yi−1−yi
sinh(βhi−1)−yi−yi+1
sinh(βhi)
−(1 −t)fi−1+tfi
γ·cosh(βhi−1)−1
sinh(βhi−1)−tfi+ (1 −t)fi+1
γ·cosh(βhi)−1
sinh(βhi)= 0,
i= 1, . . . , N −1.
ε6C
N.
xi, xi±1∈[xN/4−1, λ]∪[λ, 1/2],
i=N/4, . . . , N/2−1
cosh(βhi−1)−1
sinh(βhi−1)(y(xi−1)−y(xi)) −cosh(βhi)−1
sinh(βhi)(y(xi)−y(xi+1))
cosh(βhi−1)−1
sinh(βhi−1)+cosh(βhi)−1
sinh(βhi)
6C
N2.
ε6C
N.
xi, xi±1∈[xN/4−1, λ]∪[λ, 1/2],
y(xi−1)−y(xi)
sinh(βhi−1)−y(xi)−y(xi+1 )
sinh(βhi)
cosh(βhi−1)−1
sinh(βhi−1)+cosh(βhi)−1
sinh(βhi)
6C
N2, i =N/4, . . . , N/2−1.
ε6C
N.
xi, xi±1∈[xN/4−1, λ]∪[λ, 1/2],
γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(βhi)−1
sinh(βhi)
(1 −t)f(xi−1, y(xi−1)) + tf (xi, y (xi))
γ
·cosh(βhi−1)−1
sinh(βhi−1)−tf (xi, y(xi)) + (1 −t)f(xi+1, y(xi+1))
γ·cosh(βhi)−1
sinh(βhi)
6C
N2, i =N/4, . . . , N/2−1.
ε6C
N, ε2y00(xi) =
f(xi, y(xi)),
γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(βhi)−1
sinh(βhi)
(1 −t)f(xi−1, y(xi−1)) + tf (xi, y (xi))
γ
·cosh(βhi−1)−1
sinh(βhi−1)−tf (xi, y(xi)) + (1 −t)f(xi+1, y(xi+1))
γ·cosh(βhi)−1
sinh(βhi)
6|(1 −t)f(xi−1, y(xi−1)) + 2tf (xi, y (xi)) + (1 −t)f(xi+1, y(xi+1))|
6ε2[(1 −t) (|r00(xi−1)|+|s00(xi−1)|)
+2t(|s00(xi)|+|r00(xi)|) + (1 −t) (|s00(xi+1 )|+|r00(xi+1)|)]
6C1ε2
(1 −t)
2 + e−xi−1
ε√m
ε2+e−xi+1
ε√m
ε2
+ 2t 1 + e−xi
ε√m
ε2!
6Cε2+1
N2, i =N/4, . . . , N/2−1.
ε
max
i|y(xi)−yi|6C
ln2N/N2, i = 0, . . . , N/4−1,
1/N2, i =N/4,...,3N/4,
ln2N/N2, i = 3N/4+1, . . . , N,
y(xi)yi
xi, C > 0
N ε.
06i < N/4−1. hi−1=hihi=O(εln N/N ),
(T y)i=γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(βhi)−1
sinh(βhi)
(1 −t) cosh(βhi−1) + t
sinh(βhi−1)(y(xi−1)−y(xi))
−(1 −t) cosh(βhi) + t
sinh(βhi)(y(xi)−y(xi+1 ))
−(1 −t)f(xi−1, y(xi−1)) + tf (xi, y (xi))
γ·cosh(βhi−1)−1
sinh(βhi−1)
−tf(xi, y(xi)) + (1 −t)f(xi+1 , y(xi+1 ))
γ·cosh(βhi)−1
sinh(βhi)
=γ
2(cosh(βhi)−1) {t[y(xi−1)−2y(xi) + y(xi+1 )
−2f(xi, y(xi))
γ(cosh(βhi)−1)
+ (1 −t) [cosh(βhi) (y(xi−1)−2y(xi) + y(xi+1 ))
−f(xi−1, y(xi−1)) + f(xi+1, y(xi+1 ))
γ·(cosh(βhi)−1)
=γ
2(cosh(βhi)−1)
ty(xi−1)−2y(xi) + y(xi+1)−2ε2y00(xi)
γ(cosh(βhi)−1)
+ (1 −t) [cosh(βhi) (y(xi−1)−2y(xi) + y(xi+1 ))
−ε2y00(xi−1) + y00(xi+1)
γ·(cosh(βhi)−1).
y(xi−1)−2y(xi) + y(xi+1) = y00(xi)h2
i+y(iv)(ξ−
i) + y(iv)(ξ+
i)
24 h4
i,
y00(xi−1) + y00(xi+1)=2y00(xi) + y(iv )(η−
i) + y(iv)(η+
i)
2h2
i,
cosh(βhi) = 1 + β2h2
i
2+Oβ4h4
i,
ξ−
i∈(xi−1, xi), ξ+
i∈(xi, xi+1), η−
i∈(xi−1, xi), η+
i∈(xi, xi+1),
(T y)i=γ·t
β2h2
i+ 2O(β4h4
i)y00(xi)h2
i+y(iv)(ξ−
i) + y(iv)(ξ+
i)
24 h4
i
−2ε2y00(xi)
γβ2h2
i
2+O(β4h4
i)
+γ·(1 −t)
β2h2
i+ 2O(β4h4
i)y00(xi)h2
i+y(iv)(ξ−
i) + y(iv)(ξ+
i)
24 h4
i
·1 + β2h2
i
2+O(β4h4
i)
−ε22y00(xi) + y(iv )(η−
i)+y(iv)(η+
i)
2
γβ2h2
i
2+O(β4h4
i)
=γ·t
β2h2
i+ 2O(β4h4
i)y(iv)(ξ−
i) + y(iv)(ξ+
i)
24 h4
i−2ε2y00(xi)
γO(β4h4
i)
+γ·(1 −t)
β2h2
i+ 2O(β4h4
i)y(iv)(ξ−
i) + y(iv)(ξ+
i)
24 h4
i
+β2h2
i
2+O(β4h4
i)y00(xi)h2
i+y(iv)(ξ−
i) + y(iv)(ξ+
i)
24 h4
i
−2ε2y00(xi)
γ· O(β4h4
i)
+ε2y(iv)(η−
i) + y(iv)(η+
i)
2γh2
iβ2h2
i
2+O(β4h4
i),
|(T y)i|6Cln2N/N 2, i = 0,1, . . . , N/4−1.
N/46i < N/2.
|(T y)i|6γ
cosh(βhi−1)−1
sinh(βhi−1)+cosh(βhi)−1
sinh(βhi)
·(1 −t)
cosh(βhi−1)−1
sinh(βhi−1)(y(xi−1)−y(xi))
−cosh(βhi)−1
sinh(βhi)(y(xi)−y(xi+1 ))
+
y(xi−1)−y(xi)
sinh(βhi−1)−y(xi)−y(xi+1 )
sinh(βhi)
+
(1 −t)f(xi−1, y(xi−1)) + tf (xi, y (xi))
γ·cosh(βhi−1)−1
sinh(βhi−1)
+
tf(xi, y(xi)) + (1 −t)f(xi+1 , y(xi+1 ))
γ·cosh(βhi)−1
sinh(βhi),
|(T y)i|6C/N2, i =N/4, . . . , N/2−1.
i=N/2. hN/4−1=hN/4
s
i=N/2,
[λ, 1−λ]
P(x) =
p1(x), x ∈[x0, x1],
p2(x), x ∈[x1, x2],
pi(x), x ∈[xi−1, xi],
pN(x), x ∈[xN−1, xN],
pi(x) =
yi−yi−1
xi−xi−1
(x−xi−1) + yi−1, x ∈[xi−1, xi],
0, x /∈[xi−1, xi],
i= 1,2, . . . , N.
max
x∈[0,1] y(x)−P(x)6Cln2N/N2,
y P
[0, λ],[λ, xN/4+1] [xN/4+1 ,1/2]
[1/2,1].kP−
yk∞6kP−Pk∞+kP−yk∞,
P
P , P
(xi−1, y(xi−1)),(xi, y(xi)), i = 1,2, . . . , N ;
(xi−1, yi−1),(xi, yi+1 ), i = 1,2, . . . , N,
P(x) =
p1(x), x ∈[x0, x1],
p2(x), x ∈[x1, x2],
pi(x), x ∈[xi−1, xi],
pN(x), x ∈[xN−1, xN],
pi(x) =
yi−yi−1
xi−xi−1
(x−xi−1) + yi−1, x ∈[xi−1, xi],
0, x /∈[xi−1, xi],
i= 1,2, . . . , N.
kP−Pk∞6Cln2N/N2, x ∈[0,1].
[0, λ],
i= 1,2, . . . , N/4. hi−1=hi, hi=
O(εln N/N ). hi=O(εln N/N)
|y(x)−pi(x)|6h2
i
8max
ξ∈[xi−1,xi]|y00(ξ)|6C1
ε2ln2N
N2max
ξ∈[xi−1,xi]|s00(ξ) + r00(ξ)|
6C2
ε2ln2N
N2max
ξ∈[xi−1,xi]ε−2e−ξ
ε√m+e−(ξ−1)
ε√m+r00(ξ)
6C2
ε2ln2N
N2(ε−2+C3)6Cln2N
N2, i = 1,2, . . . , N/4.
x∈[λ, xN/4+1]∪[xN/4+1 ,1/2]
i=N/4, N/4 + 1, . . . , N/2,
i=N/4 + 1, . . . , N/2, hi=O(1/N ).
|y(x)−pi(x)|6h2
i
8max
ξ∈[xN/4+1,1/2] |y00(ξ)|6C
N2.
[λ, xN/4+ 1],
y−pi(x) =y−yi−yi−1
xi−xi−1
(x−xi−1) + yi−1
=s−si−si−1
xi−xi−1
(x−xi−1) + si−1+r−ri−ri−1
xi−xi−1
(x−xi−1) + ri−1.
s,
s−si−si−1
xi−xi−1
(x−xi−1) + si−1
6|s|+|si+1 −si|+|si|
6C1e−xi−1
ε√m+e−xi−1−1
ε√m6C
N2.
r,
r−ri−ri−1
xi−xi−1
(x−xi−1) + ri−1
6h2
i−1
8max
ξ∈[xi−1,xi]|r00(ξ)|6C
N2.
C,
C(x) = Ci(x), x ∈[xi, xi+1 ], i = 0,1, . . . , N −1,
Ci
Ci(x) = Mi
(xi+1 −x)3
6hi+1
+Mi+1
(x−xi)3
6hi+1
+yi+1 −yi
hi+1 −hi+1
6(Mi+1 −Mi)(x−xi) + yi−Mi
h2
i+1
6,
Mi:= C00
i(xi), i = 1, N −1
hi
6Mi−1+hi+hi+1
3Mi+hi+1
6Mi+1 =yi+1 −yi
hi+1 −yi−yi−1
hi
, i = 1, . . . , N −1,
M0:= C00
0(x0)=0, MN:= C00
N−1(xN) = 0.
ε2y00 =y+ cos2πx + 2ε2π2cos2πx (0,1) ,
y(0) = y(1) = 0.
y(x) = e−x
ε+e
−(1−x)
ε
1 + e−1
ε−cos2πx.
y0=−0.5
γ= 1.
EN
EN=ky−yNk∞,Ord = ln EN−ln E2N
ln(2k/(k+ 1)) ,(Shishkin),
Ord = ln EN−ln E2N
ln 2 ,(Bakhvalov,Liseikin)
N= 2k, k = 4,5,...,12, y
yN
ε
EN
x= 0
N= 16 N= 32
N= 64
2−52−10 2−20 2−30 2−40
N EnEnEnEn
24
25
26
27
28
29
210
211
212
24
25
26
27
28
29
210
211
212
24
25
26
27
28
29
210
211
212
24
25
26
27
28
29
210
211
212
EN
N= 16 N= 32
N= 64
ε
Oln2N/N 2.
C1
O numeričkom rješavanju Cauchyjevog problema Runge-Kutta metodama na Shishkinovoj mreži
Preprint
Full-text available
  • Jun 2023
In this paper, the numerical solution of the singular-perturbation Cauchy problem by Runge-Kutta methods on the Shishkin grid is discussed. Numerical solutions of the observed problem were obtained using two explicit and one implicit Runge-Kutta method on the simplest layer-adaptive network. Finally, the obtained results were compared.
View
A fitted second--order difference scheme on a modified Shishkin mesh for a semilinear singularly-perturbed boundary-value problem
Preprint
Full-text available
  • Jun 2022
In the present paper we consider the numerical solving of a semilinear singular--perturbation reaction--diffusion boundary--value problem having boundary layers. A new difference scheme is constructed, the second order of convergence on a modified Shishkin mesh is shown. The numerical experiments are included in the paper, which confirm the theoretical results.
View
Matematika 1
Preprint
Full-text available
  • Aug 2021
The book was written on the basis of materials that we presented at several faculties, either as lectures or as part of auditory exercises. Aware that there are more books and textbooks in the area in which the topics covered by this book are covered, we tried, based on the mentioned experience, to write a book oriented towards students.
View
Numerical analysis of grid-clustering rules for problems with power of the first type boundary layers
Article
Full-text available
  • Mar 2020
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. В статье приведены результаты численных расчетов обыкновенных сингулярно-возмущенных 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.
View
Compact Difference Schemes and Layer Resolving Grids for Numerical Modeling of Problems with Boundary and Interior Layers
Article
Full-text available
  • Feb 2019
A combination of two approaches to numerically solving second-order ODEs with a small parameter and singularities, such as interior and boundary layers, is considered: compact highorder approximation schemes and explicit generation of layer resolving grids. The generation of layer resolving grids, which is based on estimates of solution derivatives and formulations of coordinate transformations eliminating the solution singularities, is a generalization of a method for a firstorder scheme developed earlier. This paper presents formulas of the coordinate transformations and numerical experiments for first, second, and third order schemes on uniform and layer resolving grids for equations with boundary, interior, exponential, and power layers of various scales. Numerical experiments confirmthe uniformconvergence of the numerical solutions performedwith the compact high-order schemes on the layer resolving grids. By using transfinite interpolation or numerical solutions to the Beltrami and diffusion equations in a control metric based on coordinate transformations eliminating the solution singularities, this technology can be generalized to the solution of multidimensional equations with boundary and interior layers.
View
A robust overlapping Schwarz method for a singularly perturbed semilinear reaction-diffusion problem with multiple solutions
Article
Full-text available
  • Jan 2009
An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion two-point boundary value problem with multiple solutions. Its diffusion parameter ε2 is arbitrarily small, which induces boundary layers. The Schwarz method invokes two boundary-layer subdomains and an interior subdomain, the narrow overlapping regions being of width O(ε| ln ε|). Constructing suband super-solutions, we prove existence and investigate the accuracy of discrete solutions in particular subdomains. It is shown that when ε ≤ CN-1 and layer-adapted meshes of Bakhvalov and Shishkin types are used, one iteration is sufficient to get second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm uniformly in ε, where N is the number of mesh intervals in each subdomain. Numerical results are presented to support our theoretical conclusions.
View
Solving a nonlocal singularly perturbed problem by splines in tension
Article
Full-text available
  • Jan 1991
A numerical method for a singularly perturbed linear nonlocal problem is constructed. An exponential spline difference scheme on nonequidistant mesh is applied. An estimate of the form min(H,ε), where ε is the perturbation parameter and H is the maximal mesh step width, is obtained. Numerical experiments which demonstrate the effectiveness of the method are presented.
View
ON A FOURTH-ORDER FINITE DIFFERENCE METHOD FOR NONLINEAR TWO-POINT BOUNDARY VALUE PROBLEMS 1
Article
Full-text available
  • Jan 2003
We consider a finite difference method of order four for nonlin-ear two-point boundary value problems. In linear case the finite difference schemes lead to a tridiagonal linear system. Numerical experiments sup-port the theoretical results.
View
Approximation of singularly perturbed reaction-diffusion problems by quadratic C 1-splines
Article
Full-text available
  • Sep 2012
Collocation with quadratic C 1-splines for a singularly perturbed reaction-diffusion problem in one dimension is studied. A modified Shishkin mesh is used to resolve the layers. The resulting method is shown to be almost second order accurate in the maximum norm, uniformly in the perturbation parameter. Furthermore, a posteriori error bounds are derived for the collocation method on arbitrary meshes. These bounds are used to drive an adaptive mesh moving algorithm. Numerical results are presented.
View
Grid Generation for Problems with Boundary and Interior Layers
Book
  • Dec 2018
The book describes algorithms for generating numerical grids for solving singularly-perturbed problems by using layer-damping coordinate transformations eliminating singularities of solutions to some required order n. The layer-damping coordinate transformations are generated explicitly by certain procedures with four basic univariate locally contracting mappings. The form of any contracting function depends on the qualitative behavior of the solution. The information about the qualitative solution structure is obtained either from a theoretical analysis of simpler model equations, specifically, the ordinary differential equations discussed in Chaps. 3--5, which simulate the qualitative features of the solutions, or from a preliminary numerical calculation for similar problems on coarse grids.The theoretical analysis has revealed new forms of layer-damping functions eliminating singularities of solutions to singularly--perturbed problems and corresponding layer-resolving grids -- functions and grids above and beyond those already well known and having broad acceptance, namely, those developed by Bakhvalov and Shishkin. The grids developed by Bakhvalov and Shishkin have been applied to diverse problems, but only to problems with exponential-type layers , typically represented by functions exp⁡(−bx/εk)\exp(-bx/\varepsilon^k), occurring in problems for which the solutions of reduced (ε=0)(\varepsilon=0) problems do not have singularities. Such grids are not suitable for tackling other, wider, layers, and also require knowledge of the constant b affecting the width of the exponential layer -- when such knowledge is not always available, for example, for boundary layers in fluid-dynamics problems modeled by Navier-Stokes equations, or for interior layers in solutions to quasilinear nonautonomous problems. One spectacular example of the new layer-resolving grids being presented in the book, engendered by a function εrk/(εk+x)r\varepsilon^{rk}/(\varepsilon^k +x)^r, r>0r>0, is suitable for dealing not only with exponential layers having arbitrary widths, but with power of first type layers occurring in problems for which the solutions of reduced problems have singularities as well. Other examples of new layer--resolving grids are aimed at dealing with power of type 2 layers represented by functions (εk+x)r(\varepsilon^k +x)^r, 0<r<10< r < 1; logarithmic layers represented by functions ln⁡(εk+x)/ln⁡εk\ln(\varepsilon^k +x)/\ln\varepsilon^k ; and mixed layers. Numerical experiments are carried out in Chap. 8 for problems having various types of layers. It seems that the new layer-resolving grids described in the book should empower researchers to solve broader and more important classes of problems having not only exponential-, but power-, logarithmic-, and mixed-type boundary and interior layers.
View
View
View
Grid approximation of singularly perturbed parabolic equations with internal layers
Article
  • Jan 1988
The periodic first boundary value problem is considered in a band domain for a parabolic-type equation. Highest-order derivatives contain a parameter taking arbitrary values in the half-open interval (0,1], while the equation coefficients and free term have discontinuities of the first kind at a finite number of straight lines parallel to one of the coordinate axes. Lumped sources can also be located at these straight lines. When the parameter tends to zero there arise internal layers in the neighbourhoods of discontinuity lines for the data of the problem. To solve the boundary value problem by using grids condensing at boundary and internal layers, a difference scheme is constructed which converges uniformly in the parameter everywhere in the domain.
View