Burgers-Huxley - Purdue Universitywang838/pub/BurgersHuxley.pdf · are applied to numerically solve...
Transcript of Burgers-Huxley - Purdue Universitywang838/pub/BurgersHuxley.pdf · are applied to numerically solve...
2010 5 U & Y R + Q 32 RQ 2 �May, 2010 MATHEMATICA NUMERICA SINICA Vol.32, No.2pji]\aonced^`gbfmh_
Burgers-Huxley lk*1)W[X�SVU�YZT(i#?*Q*�, 3� 200092)K FA�#Q ε L Burgers-Huxley m2I3hqÆ#�qX+�:8Z4�m}c, �xCoQ>2(sMGk��Sh��Qm>. b_.m�L�@%, CA sinh ��LGk��ShB
Chebyshev :S[�@%8*�, p7:S!Z=I5yG^; >.m�!CoQ>2(sM 4 8 Runge-Kutta h�<�L{E, �Cr�s%XMeQLmh[ut�>�:8Z}c>PILLQm�{Xc. Qm@.��, �xa6Lmh[�>��I�@% ��%L�:8Z Burgers-Huxley }c\G5yLG^.�": oQ>2(sh; Gk��Sh; �:8Z; Burgers-Huxley }cMR (2000) P8�(: 65M70, 34E15
THE EXPONENTIAL TIME DIFFERENCING AND RATIONAL
SPECTRAL COLLOCATION METHOD FOR SINGULARLY
PERTURBED BURGERS-HUXLEY PROBLEM
Wang Yingwei Chen Suqin Wu Xionghua
(Department of Mathematics, Tongji University, Shanghai 200092, China)
Abstract
The exponential time differencing (ETD) and rational spectral collocation (RSC) method
are applied to numerically solve the Burgers-Huxley equation with small parameter ε, which
is a nonlinear, unsteady, singularly perturbed, initial and boundary problem. The RSC
method with sinh transform is employed to treat the boundary or interior layer in spatial
direction while the combination of ETD and forth-order Runge-Kutta (ETDRK4) method
is used to discretize the time variable. The matrix function in ETDRK4 is computed by
contour integral in the complex plan, which overcomes the problem of instability. Numerical
experiments illustrate the high accuracy and efficiency of our method.
Keywords: exponential time differencing method; rational spectral collocation method;
singularly perturbation; Burgers-Huxley problem
2000 Mathematics Subject Classification: 65M70, 34E15
* 2009 5 U 6 +MI.1) �(~(X*�B (NO.10671146 NO.50678122) {y��.
172 % X Q * 2010 1. G��E
Burgers-Huxley 3�l|bK2�`g-�:
∂u
∂t= ε
∂2u
∂x2− αuδ ∂u
∂x+ βu
(
1 − uδ) (
uδ − γ)
(1.1)
D ≡ {(x, t) ∈ [a, b] × (0, T ]} (1.2)
u(x, 0) = u0(x), x ∈ [a, b] (1.3)
u(a, t) = ua(t), u(b, t) = ub(t) (1.4)�t α, β, γ, δ, ε s"P, � γ ∈ (0, 1), ε > 0.E ε = 1, α, β > 0 =sHT Burgers-Huxley |b, Zydo)��)i?�;��1�}Y`sMwNF�kK?B [1]; E δ = 1, β = 0 =s Burgers |b, �\Bd�OydKayc1� [2]; E δ = 1, α = 0 =s Huxley |b, �\BK,J:H���KM�Uz [3]; E 0 < ε � 1 =s�97Y Burgers-Huxley |b (singularly perturbed Burgers-Huxley
problem),Zydo)t]\�sxfÆPf1�ydp "WY5K�?$�. �97Y Burgers-Huxley |bH2gpW*�p "K�97Y|b, wx0,Jbg|bKPl=g.Z ε = 1, "P α, β, γ, δ "_Wl6� u0(x), ua(t), ub(t) H_WPKf1�, YJHHT Burgers-Huxley |bKF%= [4]; E ε s�"P=, D�� Burgers l1F"P DK=�=, �]� [6JHF%=. Pl�=�97Y Burgers-Huxley|bFq>R: 2Hp "�K7j, -�k6aRl1E!'r�F�Q��i1, �JH>�p "BPl1�, �=]�B,[�>; eH ε ��<��?$, HTPllgK�<�. ca9|b, ~�`5s=-l�BnP=1'r 4 7 Runge-Kutta g�^-l�B@ sinh ��KFj��Rg.�lg�=�?$|bKDGZK��RHpMQ9R, Z�?7�4��. �-HTChebyshev 9R xk = cos(kπ/N): |x1 − x0| = |xN − xN−1| = |cos(π/N) − 1| ≈ 1
2
(
πN
)2≈ 5
N2 ,�q9R-QZ [−1, 1]K�?H O(N−2) 7K, Z7j�?$=, �HTF�'r, F�Q9RK O(N−1) 7FDG. D�?$_�=, HT��Rg*%4c9R. 'c)a`5swEK�lg [5, 6],�x0U�Hg���A�RZ�?$7)�. D���Kl11ZivX,^)s$Wr. Berrut [7, 8] M`5sv� DKFj��Rg (rational spectral collocation
method in barycentric form, /. RSC lg), �x0DRH<.JE���, Rprl1��Zi��. Tee M [9] `5swEK RSC-sinh lg, g�<. sinh ��, AJ���KChebshev 9RZ�?$7��. wB9lga�97Y Burgers-Huxley |bZ^-l�E!s�i1.nP=1'rg (exponential time differencing method, /. ETD lg), J.�r;|g (integration factor method, /. IF lg), H�=@=-Kprl1K2uPllg. g�al1q�h0�r;|A "�r (2�H^-K�x7GP) JHF%O7j, L "�F�KzW"�sJ6oU, ;dY6AB4>K=-�,. Cox M [10] � ETD lgL 47 Runge-Kutta g�;Æ, `5s ETDRK4 zD. 9zDY7j)7 DKp "�, DBZ�97Y|b=OHsPl�zWKb. ~� B Trefethen M [11, 12] `5Kv��q �r$WLdPKlg=SsPl�zW|b. wB ETDRK4 zD�=H^-�i
2 � m>u N: �:8Z Burgers-Huxley }c 1731�JHK�K=-K*prl1.w�,�j-�: P 2 9A6@F sinh ��K RSC lg, P 3 9}5 ETDRK4 zD6�q �r$WLdPKlg, P 4 9HPlWm, ��H�;L_o.
2. '!�C� RSC-sinh ��2.1. -��n9^-�= NH [−1, 1](/ x ∈ [a, b], ]B�� y = 2
b−a (x − a) − 1 ∈ [−1, 1]).HT Chebyshev 9R {xk = cos(kπ/N), k = 0, 1, · · · , N} Zq_R x = ±1 4�, Zt-4N. lB Tee [9] }5K��:
g(x) = ρ + ε sinh
[(
sinh−1
(
1 − ρ
ε
)
+ sinh−1
(
1 + ρ
ε
))
x − 1
2+ sinh−1
(
1 − ρ
ε
)]
. (2.1)�t, x = ρ H�?$vr, ε H�?$�] (w"sl1 (1.1) tK�"P ε).���K Chebyshev 9R {xk = g (cos(kπ/N)) , k = 0, 1, ...N} Z�?$ x = ρ 7�4��, _�JÆ7j�97Y|b.
2.2. D�:�%M6 {xk} s9RKv� DKFj&lPs [7]
rN (x) =
N∑
k=0
ωk
x−xku (xk)
N∑
k=0
ωk
x−xk
. (2.2)�t {ωk} Hv�$�P, ω0 = 12 , ωN = (−1)N
2 , ωk = (−1)k, k = 0, 1, · · · , N − 1.P rN (x) Z xj 7K n 7GPY�Fs r(n)N (xj) =
∑Nk=0 D
(n)jk u(xk), �t2�e7prLd D(1), D(2) Yg� Schneider � Werner [13] }5K�DWJ:
D(1)jk =
ωk
ωj(xj−xk) , j 6= k
−∑
i6=k
D(1)ji , j = k
(2.3)
D(2)jk =
2D(1)jk
(
D(1)jj − 1
xj−xk
)
, j 6= k
−∑
i6=j
D(2)ji , j = k
(2.4)Q 1. 2�O, /<.s�� x 7→ y = y(x), l1tKGP10E!�?K�� (�- ∂u∂x?�s ∂u
∂y = 1y′(x)
∂u∂x ). D; (2.3), (2.4) YV5, RSC lgKprLdk64eK���K9R, ;9Rl1 (1.1) ��Z�� (2.1) dw�.
174 % X Q * 2010 2.3. � ).9 U =
U1
U2
U3
, �t U1 = u(−1, t), U3 = u(1, t), U2 = [u(x1, t), ...u(xN−1, t)]
T, U1, U3H�?f1, U2 H�R7Pl�/Ku�r. x D = D(1), B = D(2) r�H2�e7prLd, �~O U KralDa D, B 1E!�?KLdra:
D =
D11 DT12 D13
D21 D22 D23
D31 DT32 D33
, B =
B11 BT12 B13
B21 B22 B23
B31 BT32 B33
.�t D11, D13, D31, D33, B11, B13, B31, B33 H?P, D12, D21, D23, D32, B12, B21, B23, B32 Hu�r, D22, B22 Hld.B RSC-sinh lgi1prl1 (1.1) J:
dU
dt= εBU − α diag(U δ)DU + f(U) (2.5)�t f(U) = βU � (1 − U δ) � (U δ − γ), U δ Hn�rK~|QV" δ :l, � �F~|QVa?�0, diag(U) �Fa3 QVs�r U Kld.2 U, D, B r�BraLd DBe, ; (2.5) t`H�Ra?Kl1:
dU2
dt= ε (B21U1 + B22U2 + B23U3) − α diag(U δ
2 ) (D21U1 + D22U2 + D23U3) + f (U2)
= εB22U2 − α diag(U δ2) (D22U2 + D21U1 + D23U3) + ε (B21U1 + B23U3) + f (U2) .(2.6)9 F = D21U1 + D23U3, G = ε (B21U1 + B23U3), ]�[YE�?f1WJ; x "W|
L(u) = εB22u, p "� N(U2) = −α diag(U δ2) (D22U2 + F ) + G + f (U2), ]F
dU2
dt= L (U2) + N (U2) . (2.7)b/~�JH2|�K=- t K*prl1�, ��2B ETDRK4 zD�=9l1�.
3. 0!�C� ETDRK4 �23.1. ���1229g���Rg�Rl1 (1.1) z�s�K=- t K*prl1 (2.7). 2�K, ~�F:
ut = Lu + N (u, t), (3.1)�t u ∈ RN H N tu�r, LHY6w�/ N 7Ld D LK "W|, N : R
N ×R 7→ RNH�K u � t Kp "W|.
ETD lgK�U�s: �al1 (3.1) q�h0�r;| e−Lt
e−Ltut − e−LtLu = e−LtN (u, t), (3.2)
2 � m>u N: �:8Z Burgers-Huxley }c 175 (
e−Ltu)
t= e−LtN (u, t). (3.3)x v = e−Ltu, ]F
vt = e−LtN (eLtv, t). (3.4)l1 (3.1) K "�r5F%O7j, '�B�hK'rlg�=l1 (3.4) KJH�hK ETDzD. w B 47 Runge-Kuttag�= (3.4), ETDRK4zD, PdWg-� [10]:x h s=-�,, 'A = h L, (3.5)
Q = heA/2 − I
A, (3.6)
f1 = h−4 − A + eA
(
4 − 3A + A2)
A3, (3.7)
f2 = h2 + A + eA (−2 + A)
A3(3.8)
f3 = h−4 − 3A − A2 + eA (4 − A)
A3, (3.9)�t I HCvd. ETDRK4 VBzDs
an = eA/2un + Q N (un, tn), (3.10)
bn = eA/2un + Q N (an, tn + h/2), (3.11)
cn = eA/2an + Q (2 N (bn, tn + h/2)− N (un, tn)) , (3.12)
un+1 = eAun + f1 N (un, tn) + 2f2 [ N (an, tn + h/2) + N (bn, tn + h/2)]
+ f3 N (cn, tn + h). (3.13)
3.2. H;B���5%M�3k66VBzD (3.10)-(3.13) �=l1 (2.7), 2OHPl�zW|b. ;s9= A =
h L = hεB22, d ε � h [��=, LdK~|QV[H��Kr, k6$WLdP (3.6)-
(3.9) �);�>K�'. ��NmT�b||b.
s(x) =
∣
∣
∣
∣
∣
e1
2x − 112x
− 1
∣
∣
∣
∣
∣
. (3.14)E��j|Yh limx→0
|s(x)| = 0. D;j 1 Y6V5, k6B�D (3.14) $W, s(x) ��LpH 0, dBq �rK$W;�L��j|�yÆ. ��2}5Bq �r$WLdPKPdlg.
Golub M [14] }5sLdPK2uW8: +9 f(z) Z2|Æ! Γ K��=�, � Γ�qsLd A K[F_fl, ]W8 f(A) -�:
f(A) =1
2π i
∮
Γ
f(z)(zI − A)−1dz, (3.15)
176 % X Q * 2010
10−15
10−10
10−5
10−14
10−12
10−10
10−8
10−6
10−4
10−2
100
x
s(x)
= |(
ex − 1
)/x
− 1
|
Explicit formulaContour integral
k 1 F x → 0 >, Q (3.14) L%X<� 5Hale M [11] n5, B�OwEK�D$WLdP�"J��K��:
f(A) =A
2π i
∮
Γ
z−1f(z)(zI − A)−1dz, (3.16)"�r! Γ s6RRsS�, �Is R KSw, ���� z = Reiπx, dz = iπzdx J:
f(A)
A=
1
2π i
∮
Γ
z−1f(z)(zI − A)−1dz
= Re
∫ 1
0
f(z)(zI − A)−1dz
= Re
M∑
j=1
∆xf(zj)(zjI − A)−1
= Re1
M
M∑
j=1
f(zj)(zjI − A)−1. (3.17)�t {zj, j = 1, . . . , M} HSw Γ 2KTV9R, ∆x = 1M , Re �F"vPK?�.��6$W Q sm, T�B�D (3.17) $WLdPKlg." f(A) = e
A2 − I, E (3.6)�(3.17) J:
Q = hf(A)
A=
h
MRe
M∑
j=1
(
ezj/2 − 1)
(zjI − A)−1. (3.18)
4. 4N7,* 1. W{F��?$K Burgers |b:
ut = εuxx − uux, x ∈ [0, 1], t > 0, (4.1)
u(x, 0) = −sin(πx), (4.2)
u(0, t) = u(1, t) = 0. (4.3)
2 � m>u N: �:8Z Burgers-Huxley }c 177� 1 � ε = 1e − 2, t = 1 /, ETDRSC �� 6+ 1 Æ>�>.�- ∆t _.SQ N L2 �( L∞ �(1e-2
16 1.7485e-3 1.1376e-3
32 7.0572e-5 2.3346e-5
64 9.9747e-5 2.3258e-5
1e-3
16 1.7528e-3 1.1335e-3
32 7.0400e-5 2.2979e-5
64 9.9568e-5 2.2960e-5� 2 ?I<@O+ 1 Æ 6#�mh >. _.SQ �(SRDQ [15] ∆t = 1e − 4, t = 0.3 30 1.96e-3 (L1)
EEFD [16] ∆t = 1e − 4, t = 1.0 80 4.18e-3 (L∞)
BSQI [17] ∆t = 1e − 3, t = 1.0 100 1.00e-3 (L∞)
CFD6 [18] ∆t = 1e − 4, t = 1.0 100 5.00e-5 (L∞)]F"P DK=�=u(x, t) =
4πε∞∑
n=1nane−εn2π2t sin(nπx)
a0 + 2∞∑
n=1ane−εn2π2t cos(nπx)
. (4.4)�t an = (−1)nIn
(
12επ
)
, In(z) �FP2g$gK Bessel P.� 1 �� 2 [H ε = 1e − 2, =m 1 K�=F]. � 1 Hw`5K ETDRSC lgZ"�h=-�,�^-RP=JHK�'. Mansell [15] B Nrvpr��g (split range
differential quadrature, /. SRPDQ)�Kutluay [16] BF%F'rg (Exact-explicit finite
difference,/. EEFD)�Zhu [17] B0: B/f&lg (B-spline quasi-interpolant,/. BSQI)� Sari [18] B 6 7C'rzD (sixth-order compact finite difference, /. CFD6) K$W;�Z� 2 t}5. Y6VH, wKlgZ ∆t = 1e − 2, N = 32 =5H<H 10−5 KF], d�\lgZ ∆t < 1e − 4 � N > 100 =�F�hKF].j 2 HPl=LF%=K�4. E ε < 1e − 2 =, "P (4.4) �Lp, D~�Klg*Y6}5Pl= (-j 3).Q 2. w� [16]�[17]�[18]t�=Kl1��?f1L (4.1)�(4.3) H2qK, D�3Cf1 (4.2) ws u(x, 0) = sin(πx).* 2. W{F��$K Burgers |but = εuxx + uux, x ∈ [−1, 1], t > 1. (4.5)("JEK3�lf1, A�F%=s
u(x, t) =sinh
(
x2ε
)
exp(
− t4ε
)
+ cosh(
x2ε
) . (4.6)
178 % X Q * 2010
0 0.5 1−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0t = 0.1
x
u
0 0.5 1−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0t = 0.3
xu
0 0.5 1−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0t = 0.6
x
u
0 0.5 1−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0t = 1.0
x
u
Exact solutionNumerical solutionk 2 ε = 1e − 2 >n 1 LQm> G&> (∆t = 1e − 3, N = 64)
0 0.2 0.4 0.6 0.8 1−1
−0.9
−0.8
−0.7
−0.6
−0.5
−0.4
−0.3
−0.2
−0.1
0
x
u
ε = 1e−4
t = 0t = 0.3t = 0.6t = 0.9k 3 n 1 LQm> (ε = 1e − 4, ∆t = 1e − 4, N = 128)� 3 }5s ε = 7.5e− 4, t = 2 ="�h=-�,�^-RJHK�'. j 4 HPl=LF%=K�4.* 3. W{FH�?$K�97Y Burgers-Huxley |b
ut = εuxx − αuδux + βu(
1 − uδ) (
uδ − γ)
, x ∈ [0, 1], t > 0, (4.7)
u(x, 0) = sin(πx), (4.8)
u(0, t) = u(1, t) = 0. (4.9)]}FDKF%=.
KaushikM [19] Z=-l�B Euler=zD, Z^-l�Br`2qnzK@tF�'rlg}5s9|bZ δ = 1 =KPl=. ~�" N = 50, ∆t = 1e − 3 JHPl=. j 5 �j 6 Lw� [19] t" N = 128, ∆t = 1e − 4 JHKPl=H2qK. ;tYV5, ε �^�
2 � m>u N: �:8Z Burgers-Huxley }c 179� 3 � ε = 7.5e − 4, t = 2 /, ETDRSC �� 6+ 2 Æ>�>.�- ∆t _.SQ N 2- jQ�( ��jQ�(1e-2
20 1.6137e-3 5.5012e-4
30 1.5998e-4 4.7572e-5
40 1.0042e-4 5.5272e-5
1e-3
20 1.6152e-3 5.5025e-4
30 1.5337e-4 4.7002e-5
40 5.8561e-5 2.2486e-5
−1 −0.5 0 0.5 1−1.5
−1
−0.5
0
0.5
1
1.5
x
u
ε = 7.5e−4, t = 2
Numerical solutionExact solution
k 4 ε = 7.5e − 4, t = 2 >n 2 LQm> G&> (∆t = 1e − 3, N = 50)
(ε = 2−1) =, 9l1K=6Z=-S0sx; E ε s�"P (ε = 2−6) =, 9l1K=� /�?$, YS0H 0, d� ε T�, �?$T�.j 7–j 10r�}5s"P α, β, γ, δ, ε��==K�b, bH�\w�}F}5K.;tYVH, α T>, �?$ /T\; β, γ, δ al1=KA�Z=-Kf_dc>.
0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
u
t = 0.1
ε = 2−2
ε = 2−4
ε = 2−8
ε = 2−16
0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
u
t = 0.9
ε = 2−1
ε = 2−3
ε = 2−6
ε = 2−9
k 5 ε ��>n 3 LQm> (α = 1, β = 1, γ = 0.5, δ = 1)
180 % X Q * 2010
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
u
ε = 2−1
t = 0.1t = 0.3t = 0.6t = 0.9
0 0.5 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x
u
ε = 2−6
t = 0.1t = 0.3t = 0.6t = 0.9
k 6 �i>]Ln 3 LQm> (α = 1, β = 1, γ = 0.5, δ = 1, ε = 2−1, 2−6)
0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
u
t = 0.1
α = 0.1
α = 1
α = 2
α = 4
0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
u
t = 0.9
α = 0.1
α = 1
α = 2
α = 4
k 7 α ��>n 3 LQm> (β = 1, γ = 0.5, δ = 1, ε = 2−8)
0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
u
t = 0.1
β = 0.1
β = 1
β = 5
β = 10
0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
u
t = 0.9
β = 0.1
β = 1
β = 5
β = 10
k 8 β ��>n 3 LQm> (α = 1, γ = 0.5, δ = 1, ε = 2−8)
2 � m>u N: �:8Z Burgers-Huxley }c 181
0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
u
t = 0.1
γ = 0.01
γ = 0.1
γ = 0.5
γ = 0.99
0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
u
t = 0.9
γ = 0.01
γ = 0.1
γ = 0.5
γ = 0.99
k 9 γ ��>n 3 LQm> (α = 1, β = 1, δ = 1, ε = 2−8)
0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
u
t = 0.1
δ = 0.5δ = 1δ = 2δ = 4
0 0.5 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
ut = 0.9
δ = 0.5δ = 1δ = 2δ = 4
k 10 δ ��>n 3 LQm> (α = 1, β = 1, γ = 0.5, ε = 2−8)
5. R$JL9w�@F sinh ��K��Rg�nP=1'r 4 7 Runge-Kutta gLq �r$WLdPKlg�;Æ, `5s�97Y Burgers-Huxley l1�KPl=g. PlK-��, lg"4>K=-�,�44K^-�R Y<H4xKF], YF�O7j�97YBurgers-Huxley |b. ~�2�bulgl�Hc|�? (�) $�vrZ=-��K�$|b�xt� . � & = A[1] Ablowitz M, Fuchssteiner B, Kruskal M. Topics in Soliton Theory and Exactly Solvable Nonlinear
Equations[M]. World Scientific, Singapore, 1987.
[2] Burgers J M. A mathematical model illustrating the theory of turbulence[J]. Adv. Appl. Mech.,
1948, 1: 171-199.
182 % X Q * 2010 [3] Huxley A F. Muscle stvucture and theories of contraction[J]. Prog Biophysics and Biophysical
Chemistry, 1957, 7: 255-318.
[4] Yefimova O Yu, Kudryashov N A. Exact solutions of the Burgers-Huxley equation[J]. Journal of
Applied Mathematics and Mechanics, 2004, 68: 413-420.
[5] Tang Tao, Manfred R Trummer. Boundary layer resolving pseudospectral methods for singular
perturbation problems[J]. SIAM J. Sci. Comp., 1996, 17: 430-438.
[6] Mulholland L S, Huang W, Solan D M. Pseudospectral solution of near-singular problems using
numerical coordinate transformations based on adaptivity[J]. SIAM J. Sci. Comput., 1998, 19:
1261-1289.
[7] Baltensperger R, Berrut J P, Noel B. Expeonential convergence of a linear rational interpolant
between transformed Chebyshev points[J]. Math. Comput., 1999, 68: 1109-1120.
[8] Berrut J P, Trefethen L N. Barycentric Lagrange interpolation[J]. SIAM Rev., 2004, 46: 501-517.
[9] Tee T W, Trefethen L N. A rational spectral collocation method with adaptively transformed
Chebyshev grid points[J]. SIAM J. Sci. Comp., 2006, 28: 1798-1811.
[10] Cox S M, Matthews P C. Exponential time differencing for stiff systems[J]. J. Comput. Phys.,
2002, 176: 430-455.
[11] Hale N, Higham N J, Trefethen L N. Computing Aα, log(A) and Related Matrix Functions by
Contour Integrals[J]. SIAM J. Numer. Anal. 2008, 46: 2505-2523.
[12] Kassam A and Trefethen L N. Fourth-order time-stepping for stiff PDEs[J]. SIAM J. Sci. Comput.,
2003, 26: 1214-1233.
[13] Schneider C, Werner W. Some new aspects of rational interpolation[J]. Mathematics of Compu-
tation, 1986, 47: 285-299.
[14] Golub G H, Van Loan C F. Matrix Computations[M] 3rd Edition. The John Hopkins University
Press,Baltimore, 1996.
[15] Mansell G, Merryfield W, Shizgal B. A comparison of differential quadrature methods for the solu-
tion of partial differential equations[J]. Computer Methods in Applied Mechanics and Engineering,
1993, 104: 259-316.
[16] Kutluay S, Bahadir A R, Ozdes A. Numerical solution of one-dimensional Burgers equation:
explicit and exact-explicit finite difference methods[J]. Journal of Computational and Applied
Mathematics, 1999, 103: 251-261.
[17] Zhu Chun-Gang, Wang Ren-Hong. Numerical solution of Burgers equation by cubic B-spline
quasi-interpolation[J]. Applied Mathematics and Computation, 2009, 208: 260-272.
[18] Murat Sari, Gurhan Gurarslan. A sixth-order compact finite difference scheme to the numerical
solutions of Burgers’ equation[J]. Applied Mathematics and Computation, 2009, 208: 475-483.
[19] Aditya Kaushik, Sharma M D. A uniformly convergent numerical method on non-uniform mesh
for singularly perturbed unsteady Burger-Huxley equation[J]. Applied Mathematics and Compu-
tation, 2008, 195: 688-706.