Burgers-Huxley - Purdue Universitywang838/pub/BurgersHuxley.pdf · are applied to numerically solve...

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2010 5 32 2 May, 2010 MATHEMATICA NUMERICA SINICA Vol.32, No.2 Burgers-Huxley *1) ( , 200092) ε Burgers-Huxley , . , sinh Chebyshev , ; 4 Runge-Kutta , . , Burgers-Huxley . : ; ; ; Burgers-Huxley MR (2000) : 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 6 . 1) (NO.10671146 NO.50678122) .

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*�, p&#55: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

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explicit and exact-explicit finite difference methods[J]. Journal of Computational and Applied

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