Du Fort–Frankel finite difference scheme for Burgers equation

Arabian Journal of Mathematics, Oct 2012

In this paper we apply the Du Fort–Frankel finite difference scheme on Burgers equation and solve 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 10−4 and observe that the numerical solutions are in good agreement with the exact solution. Open image in new window

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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)


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K. Pandey, Lajja Verma, Amit K. Verma. Du Fort–Frankel finite difference scheme for Burgers equation, Arabian Journal of Mathematics, 2012, pp. 91-101, Volume 2, Issue 1, DOI: 10.1007/s40065-012-0050-1