A Note on Crank-Nicolson Scheme for Burgers’ Equation

ABSTRACT

In this work we generate the numerical solutions of the Burgers’ equation by applying the Crank-Nicolson method directly to the Burgers’ equation, i.e., we do not use Hopf-Cole transformation to reduce Burgers’ equation into the linear heat equation. Absolute error of the present method is compared to the absolute error of the two existing methods for two test problems. The method is also analyzed for a third test problem, nu-merical solutions as well as exact solutions for different values of viscosity are calculated and we find that the numerical solutions are very close to exact solution.

In this work we generate the numerical solutions of the Burgers’ equation by applying the Crank-Nicolson method directly to the Burgers’ equation, i.e., we do not use Hopf-Cole transformation to reduce Burgers’ equation into the linear heat equation. Absolute error of the present method is compared to the absolute error of the two existing methods for two test problems. The method is also analyzed for a third test problem, nu-merical solutions as well as exact solutions for different values of viscosity are calculated and we find that the numerical solutions are very close to exact solution.

Cite this paper

nullK. Pandey and L. Verma, "A Note on Crank-Nicolson Scheme for Burgers’ Equation,"*Applied Mathematics*, Vol. 2 No. 7, 2011, pp. 883-889. doi: 10.4236/am.2011.27118.

nullK. Pandey and L. Verma, "A Note on Crank-Nicolson Scheme for Burgers’ Equation,"

References

[1] N. Su, J. P. C. Watt, K. W. Vincent, M. E. Close and R. Mao, “Analysis of Turbulent Flow Patterns of Soil Water under Field Conditions Using Burgers’ Equation and Porous Suction-Cup Samplers,” Australian Journal of Soil Research, Vol. 42, No. 1, 2004, pp. 9-16. doi:10.1071/SR02142

[2] N. J. Zabusky and M. D. Kruskal, “Interaction of Solitons in a Collisionless Plasma and the Recurrence of Initial States,” Physical Review, Vol. 15, No. 6, 1965, pp. 240-243. doi:10.1088/0305-4470/33/18/308

[3] P. F. Zhao and M. Z. Qin, “Multisymplectic Geometry and Multisymplectic Preissmann Scheme for the Kdv Equation,” Journal of Physics A, Vol. 33, No. 18, 2000, pp. 3613-3626.

[4] H. Brezis and F. Browder, “Partial Differential Equations in the 20th Century,” Advances in Mathematics, Vol. 135, No. 1, 1998, pp. 76-144. doi:10.1006/aima.1997.1713

[5] J. D. Cole, “On a Quasilinear Parabolic Equation Occurring in Aerodynamics,” Quarterly of Applied Mathematics, Vol. 9, 1951, pp. 225-236.

[6] E. Hopf, “The Partial Differential Eqaution ,” Communications on Pure and Applied Mathematics, Vol. 3, 1950, pp. 201-230. doi:10.1002/cpa.3160030302

[7] M. K. Kadalbajoo and A. Awasthi, “A Numerical Method Based on Crank-Nicolson Scheme for Burgers’ Equation,” Applied Mathematics and Computation, Vol. 182, No. 2, 2006, pp. 1430-1442. doi:10.1016/j.amc.2006.05.030

[8] A. Gorguis, “A Comparison between Cole-Hopf Transformation and Decomposition Method for Solving Burgers’ Equations,” Applied Mathematics and Computation, Vol. 173, No. 1, 2006, pp. 126-136. doi:10.1016/j.amc.2005.02.045

[9] S. Kutluay, A. Esen and I. Dag, “Numerical Solutions of the Burgers’ Equation by the Least—Squares Quadratic B-Spline Finite Element Method,” Journal of Computational and Applied Mathematics, Vol. 167, No. 1, 2004, pp. 21-33. doi:10.1016/j.cam.2003.09.043

[10] K. Pandey, L. Verma and A. K. Verma, “On a Finite Difference Scheme for Burgers’ Equation,” Applied Mathematics and Computation, Vol. 215, No. 6, 2009, pp. 2206-2214. doi:10.1016/j.amc.2009.08.018

[11] M. K. Jain, “Numerical Solution of Differential Equations,” New Age International (P) Limited, New Delhi, 1984.

[12] G. D. Smith, “Numerical Solution of Partial Differential Equations,” Oxford University Press, Oxford, 1978.

[1] N. Su, J. P. C. Watt, K. W. Vincent, M. E. Close and R. Mao, “Analysis of Turbulent Flow Patterns of Soil Water under Field Conditions Using Burgers’ Equation and Porous Suction-Cup Samplers,” Australian Journal of Soil Research, Vol. 42, No. 1, 2004, pp. 9-16. doi:10.1071/SR02142

[2] N. J. Zabusky and M. D. Kruskal, “Interaction of Solitons in a Collisionless Plasma and the Recurrence of Initial States,” Physical Review, Vol. 15, No. 6, 1965, pp. 240-243. doi:10.1088/0305-4470/33/18/308

[3] P. F. Zhao and M. Z. Qin, “Multisymplectic Geometry and Multisymplectic Preissmann Scheme for the Kdv Equation,” Journal of Physics A, Vol. 33, No. 18, 2000, pp. 3613-3626.

[4] H. Brezis and F. Browder, “Partial Differential Equations in the 20th Century,” Advances in Mathematics, Vol. 135, No. 1, 1998, pp. 76-144. doi:10.1006/aima.1997.1713

[5] J. D. Cole, “On a Quasilinear Parabolic Equation Occurring in Aerodynamics,” Quarterly of Applied Mathematics, Vol. 9, 1951, pp. 225-236.

[6] E. Hopf, “The Partial Differential Eqaution ,” Communications on Pure and Applied Mathematics, Vol. 3, 1950, pp. 201-230. doi:10.1002/cpa.3160030302

[7] M. K. Kadalbajoo and A. Awasthi, “A Numerical Method Based on Crank-Nicolson Scheme for Burgers’ Equation,” Applied Mathematics and Computation, Vol. 182, No. 2, 2006, pp. 1430-1442. doi:10.1016/j.amc.2006.05.030

[8] A. Gorguis, “A Comparison between Cole-Hopf Transformation and Decomposition Method for Solving Burgers’ Equations,” Applied Mathematics and Computation, Vol. 173, No. 1, 2006, pp. 126-136. doi:10.1016/j.amc.2005.02.045

[9] S. Kutluay, A. Esen and I. Dag, “Numerical Solutions of the Burgers’ Equation by the Least—Squares Quadratic B-Spline Finite Element Method,” Journal of Computational and Applied Mathematics, Vol. 167, No. 1, 2004, pp. 21-33. doi:10.1016/j.cam.2003.09.043

[10] K. Pandey, L. Verma and A. K. Verma, “On a Finite Difference Scheme for Burgers’ Equation,” Applied Mathematics and Computation, Vol. 215, No. 6, 2009, pp. 2206-2214. doi:10.1016/j.amc.2009.08.018

[11] M. K. Jain, “Numerical Solution of Differential Equations,” New Age International (P) Limited, New Delhi, 1984.

[12] G. D. Smith, “Numerical Solution of Partial Differential Equations,” Oxford University Press, Oxford, 1978.