On the Convergence and Stability of Finite Difference Method for Parabolic Partial Differential Equations
Journal of Advances in Mathematics and Computer Science,
Page 58-67
DOI:
10.9734/jamcs/2021/v36i1030412
Abstract
In this paper, we verify the convergence and stability of implicit (modified) finite difference scheme. Knowing fully that consistency and stability are very important criteria for convergence, we have prove the stability of the modied implicit scheme using the von Newmann method and also verify the convergence by comparing the numerical solution with the exact solution. The results shows that the schemes converges even as the step size is rened.
Keywords:
- Finite difference scheme
- Crank-Nicolson scheme
- stability
- modifed Crank-Nicolson scheme
- diffusion equations.
How to Cite
Omowo, B. J., Longe, I. O., Abhulimen, C. E., & Oduwole, H. K. (2021). On the Convergence and Stability of Finite Difference Method for Parabolic Partial Differential Equations. Journal of Advances in Mathematics and Computer Science, 36(10), 58-67. https://doi.org/10.9734/jamcs/2021/v36i1030412
References
Abhulimen CE, Omowo BJ. Modied Crank-Nicolson method for solving one dimensional parabolic equation. International Journal of Scientic Research. 2019;15(6)series 3:60-66
Crank J, Philis N. A practical method for Numerical Evaluation of solution of partial dierential equation of heat conduction type. Proc. camb. Phil. Soc. 1996;1:50-57.
Omowo BJ, Abhulimen CE. On the stability of Modied Crank-Nicolson method for Parabolic Partial dierential equations. International Journal of Mathematical Sciences and Optimization: Theory and Application. 2021;6(2):862-873.
Omowo Babajide Johnson, Longe Idowu Oluwaseun. Crank-Nicolson and Modied Crank- Nicolson scheme for one dimensional parabolic partial dierential equation. International journal of Applied Mathematics and Theoretical Physics. 2020;6(3):35-40.
Febi Sanjaya, Sudi Mungkasi. A simple but accurate explicit nite dierence method for Advection-diusion equation. Journal of Phy. Conference Series 909; 2017.
Qiqi Tran, Jinjie Lin. Modied Iterated Crank-Nicolson method with improved Accuracy. arXiv: 1608.01344 V1 [math.NA].
Simeon Kiprono Mariton. Modied Crank Nicholson Based Methods on the Solution of one dimensional Heat Equation. Nonlinear Analysis and Dierential Equations. 2019;7(1):33-37.
Cooper J. Introduction to Partial dierential Equation with Matlab. Boston; 1958.
Mitchell AR, Gridths DF. A Finite dierence method in partial dierential equations. John Wiley and Sons; 1980.
Williams F. Ames, Numerical methods for Partial dierential Equations, Academic Press, Inc, Third Edition; 1992.
Smith GD. Numerical solution of partial dierential equations: Finite dierence methods. Clarendon Press, Third Edition, Oxford; 1985.
Grewal BS. Higher Engineering Mathematics. Khanna Publisher, Forty-second Edition; 2012.
John Strikwerda. Finite dierence schemes and Partial dierential equations. SIAM, Society for Industrial and Applied Mathematics; 2004.
Crank J, Philis N. A practical method for Numerical Evaluation of solution of partial dierential equation of heat conduction type. Proc. camb. Phil. Soc. 1996;1:50-57.
Omowo BJ, Abhulimen CE. On the stability of Modied Crank-Nicolson method for Parabolic Partial dierential equations. International Journal of Mathematical Sciences and Optimization: Theory and Application. 2021;6(2):862-873.
Omowo Babajide Johnson, Longe Idowu Oluwaseun. Crank-Nicolson and Modied Crank- Nicolson scheme for one dimensional parabolic partial dierential equation. International journal of Applied Mathematics and Theoretical Physics. 2020;6(3):35-40.
Febi Sanjaya, Sudi Mungkasi. A simple but accurate explicit nite dierence method for Advection-diusion equation. Journal of Phy. Conference Series 909; 2017.
Qiqi Tran, Jinjie Lin. Modied Iterated Crank-Nicolson method with improved Accuracy. arXiv: 1608.01344 V1 [math.NA].
Simeon Kiprono Mariton. Modied Crank Nicholson Based Methods on the Solution of one dimensional Heat Equation. Nonlinear Analysis and Dierential Equations. 2019;7(1):33-37.
Cooper J. Introduction to Partial dierential Equation with Matlab. Boston; 1958.
Mitchell AR, Gridths DF. A Finite dierence method in partial dierential equations. John Wiley and Sons; 1980.
Williams F. Ames, Numerical methods for Partial dierential Equations, Academic Press, Inc, Third Edition; 1992.
Smith GD. Numerical solution of partial dierential equations: Finite dierence methods. Clarendon Press, Third Edition, Oxford; 1985.
Grewal BS. Higher Engineering Mathematics. Khanna Publisher, Forty-second Edition; 2012.
John Strikwerda. Finite dierence schemes and Partial dierential equations. SIAM, Society for Industrial and Applied Mathematics; 2004.
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