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Du Fort–Frankel finite difference scheme for Burgers equation
K. Pandey
0
Lajja Verma
0
Amit K. Verma
0
0
K. Pandey Department of Mathematics and Astronomy, University of Lucknow
, Lucknow 226007,
India
three test problems. We calculate the numerical solutions using Mathematica 7.0 for different values of viscosity. We have considered smallest value of viscosity as 104 and observe that the numerical solutions are in good agreement with the exact solution. Mathematics Subject Classification 65N06 (Du Fort-Frankel) Mathematica 7.0
-
t
(x , t ) (0, 1) (0, T ]
w(0, t ) = g1(t ),
2 Exact solution
with Fourier coefficients as
(x , t ) = A0 +
An exp
n=1
A0 =
exp
An = 2
exp
4 Numerical results and discussion
4.1 Problem 1
3 Description of the method
1
A0 =
2x 2
(1 cos x ) dx , An = 2
4.2 Problem 2
1
2x 2
(3 2x ) dx , An = 2
(3 2x ) cos(n x )dx .
h = 0.025, d = 0.125
h = 0.0125, d = 0.03125
Exact solution
Computed solution
Exact solution
Computed solution
Exact solution
h = 0.025, d = 0.125
Exact solution
Computed solution
Exact solution
Computed solution
Exact solution
h = 0.05, d = 0.5
Computed solution
h = 0.025, d = 0.125
Exact solution
Computed solution
Exact solution
Computed solution
Exact solution
0.307354
0.598069
0.853758
1.05295
1.1709
1.18163
1.0638
0.810417
0.439768
0.235166
0.459645
0.661882
0.828473
0.942984
0.984667
0.928516
0.766497
0.284701
0.263835
0.500182
0.698449
0.851921
0.952114
0.98746
0.941272
0.783701
0.548686
Computed solution
Exact solution
Computed solution
Exact solution
Computed solution
Exact solution
Problem 4.1
0.109517
0.209758
0.291865
0.34791
0.371591
0.359088
0.309965
0.227876
0.120722
h = 0.05, d = 0.5
Computed solution
0.201986
0.392435
0.559111
0.68847
0.765389
0.773827
0.699498
0.535755
0.292094
h = 0.05, d = 0.5
Computed solution
0.211009
0.408366
0.578747
0.709105
0.786093
0.795418
0.722576
0.557692
0.306233
0.39334
0.785588
1.17496
1.55788
1.92636
2.26047
2.50327
2.48185
1.76276
0.10954
0.20979
0.2919
0.34792
0.37158
0.35905
0.30991
0.22782
0.12069
0.20241
0.393201
0.560073
0.689456
0.76625
0.774471
0.699912
0.535983
0.292192
0.211315
0.408941
0.579501
0.709887
0.786732
0.795784
0.722638
0.557546
0.306075
0.394264
0.787021
1.17609
1.55755
1.92321
2.2532
2.49142
2.4675
1.75224
Problem 4.2
0.112918
0.216311
0.301055
0.358998
0.383589
0.370858
0.320261
0.235537
0.12481
0.228169
0.445803
0.641386
0.801371
0.909044
0.943621
0.880778
0.698132
0.391534
0.248593
0.477793
0.673289
0.824944
0.92286
0.954344
0.901529
0.738341
0.43355
0.395957
0.793275
1.19301
1.59555
2.00002
2.4034
2.79839
3.16762
3.42563
0.11289
0.21625
0.30097
0.35886
0.38342
0.37066
0.32007
0.23537
0.12472
0.228675
0.446428
0.641476
0.800237
0.906278
0.939401
0.876009
0.694143
0.389365
0.24903
0.478258
0.673046
0.823281
0.919457
0.949488
0.89614
0.733728
0.430905
0.396695
0.793091
1.18862
1.58211
1.97127
2.35182
2.71544
3.04407
3.25473
Problem 4.3
0.306694
0.596984
0.852678
1.05241
1.17139
1.18336
1.06648
0.813219
0.441575
0.234829
0.460554
0.666655
0.839693
0.961708
1.00881
0.951415
0.760721
0.428993
0.263585
0.501686
0.704549
0.86484
0.972233
1.01244
0.966159
0.807204
0.497518
0.396928
0.800338
1.21617
1.64929
2.10304
2.5787
3.07465
3.58445
4.09173
Numerical solutions
N = 10
0.00969751
0.0194008
0.0291023
0.0388169
0.0485068
0.0581065
0.0670385
0.0722915
0.0592958
N = 20
0.00969206
0.0193849
0.0290791
0.0387738
0.0484601
0.058076
0.0672139
0.0732623
0.0618627
Numerical solutions
N = 20
0.00974103
0.0194827
0.0292253
0.0389682
0.0487025
0.0583674
0.0675619
0.0736923
0.0623514
N = 20
0.00986103
0.0197225
0.0295845
0.0394462
0.0492993
0.0590865
0.0684217
0.0747558
0.0635649
N = 20
0.00100288
0.00200579
0.00300874
0.00401165
0.00501376
0.00600971
0.00696259
0.00762217
0.00651922
N = 10
0.0097647
0.0195351
0.0293001
0.0390783
0.0488257
0.0584871
0.067481
0.0728243
0.0598502
N = 10
0.00993853
0.0198835
0.0298153
0.039764
0.0496677
0.0595009
0.068669
0.0742707
0.0613733
N = 10
0.0010008
0.00200233
0.00300231
0.00400428
0.0050013
0.00599212
0.00691669
0.00748855
0.00620309
Numerical solutions
Numerical solutions
Exact solution
0.00965798
0.0193482
0.0290787
0.0388117
0.0484579
0.0579005
0.0669137
0.073627
0.0641923
Exact solution
0.00973453
0.019469
0.0292031
0.0389358
0.0486598
0.058324
0.0675733
0.0739831
0.0632856
Exact solution
0.00984981
0.0197005
0.0295514
0.0394
0.0492395
0.0590263
0.0684243
0.0750494
0.064521
Exact solution
0.000993238
0.00199047
0.00299277
0.00399563
0.00498799
0.00595643
0.00688528
0.00762049
0.00678091
N = 40
0.00969306
0.0193867
0.0290812
0.0387761
0.0484638
0.0580905
0.0672847
0.0735637
0.0626532
N = 40
0.00974118
0.0194828
0.0292254
0.0389681
0.0487038
0.05838
0.0676319
0.0739968
0.0631584
N = 40
0.00985735
0.0197152
0.029574
0.0394332
0.0492864
0.0590844
0.0684784
0.0750563
0.0644039
N = 40
0.00100595
0.00201193
0.00301794
0.00402393
0.00502925
0.00602928
0.00699038
0.00767449
0.0066196
N = 80
0.0096873
0.0193746
0.0290618
0.0387478
0.0484249
0.0580402
0.067229
0.0735353
0.0627219
N = 80
0.0097353
0.0194723
0.0292125
0.03 (...truncated)