Journal of Advances in Applied Mathematics

On the Convergence of Non-Polynomial Spline Finite Difference Method for Quasi-Linear Elliptic Boundary Value Problems in Two-Space Dimensions

Download PDF (1186.4 KB) PP. 59 - 72 Pub. Date: January 1, 2016

DOI: 10.22606/jaam.2016.11006

Author(s)

Navnit Jha^*
Department of Mathematics, South Asian University, Chanakyapuri, New Delhi 110 021, India
Ravindra Kumar
Department of Mathematics, Rajdhani College, University of Delhi, Delhi 110 015, India
R. K. Mohanty
Department of Mathematics, Rajdhani College, University of Delhi, Delhi 110 015, India.

Abstract

An approximate solution technique based on non-polynomial spline and finite difference method has been described for two-space dimensional quasi-linear elliptic boundary value problems. The numerical scheme in the limiting case of non-polynomial spline parameter provides the cubic spline scheme. The proposed scheme is analyzed for the convergence using matrix theory. Experimental results show the importance of using non-polynomial spline scheme over corresponding cubic spline scheme in terms of iterations to achieve desired accuracy. The method is tested for the convergence and corroborates the theoretical truncation errors. The accuracy in the solutions is obtained for Tricomi and Khokhlov- Zabolotskaya equations for various mesh step sizes.

Keywords

Finite difference method, Non-polynomial spline, Tricomi equation, Khokhlov-Zabolotskaya equation, Maximum absolute errors.

References

[1] Andrew, V.K., Kaptsov, O.V., Pukhnachov, V.V., Rodionov, A.A. Applications of group theoretical methods in hydrodynamics. Kulwer, Dordrecht (1999)

[2] Aziz, I., Siraj-ul-Islam, ?arler, B. “Wavelets collocation methods for the numerical solution of elliptic BV problems.” Appl. Math. Model. 37, 676--694 (2013)

[3] Babich, V.M., Kapilevich, M.B., Mikhlin, S.G. Linear equations of mathematical physics. Nauka, Moscow (1964)

[4] Deka, B. “Finite element methods with numerical quadrature for elliptic problems with smooth interfaces.” J. Comput. Appl. Math. 234, 605--612 (2010)

[5] Evans, L.C. Partial differential equations. American Mathematical Society Providence, Rhode Island (2002)

[6] Gopal, V., Mohanty, R.K., Saha, L.M. “A new high accuracy non-polynomial tension spline method for the solution of one dimensional wave equation in polar co-ordinates.” J. Egyptian Math. Soc. 22, 280--285 (2014)

[7] Hageman, L.A., Young, D.M. Applied iterative methods. Dover Publication, New York (2004)

[8] Henrici, P. Discrete variable methods in ordinary differential equations. Wiley, New York (1962)

[9] Hu, H.Y. “Radial basis collocation methods for elliptic boundary value problem.” Comput. Math. Appl. 50, 289--320 (2005)

[10] Ibragimov, N.H. Transformation groups applied in mathematical physics. D. Reidel Publ., Dordrecht (1985)

[11] Jha, N., Mohanty, R.K. “Quintic hyperbolic non-polynomial spline and finite difference method for nonlinear second order differential equations and its application.” J. Egyptian Math. Soc. 22, 115--122 (2014)

[12] Jha, N. “High order accurate quintic non-polynomial spline finite difference approximations for the numerical solution of non-linear two point boundary value problems.” Int. J. Model. Simul. Sci. Comput. 5, 1--16 (2014)

[13] Khan, A. “Parametric cubic spline solution of two point boundary value problems.” Appl. Math. Comput. 154, 175--182 (2004)

[14] Kozlov, V.V. Symmetries, topology and resonances in Hamiltonian mechanics. Izd-vo Udmurtskogo Gos. Universiteta, Izhevsk (1995)

[15] Lambert, J.D. Computational methods in ordinary differential equations. Wiley (1973)

[16] Lynch, R.E., Rice, J.R. “High accuracy finite difference approximation to solutions of elliptic partial differential equations.” Proc. Natl. Acad. Sci. 75, 2541--2544 (1978)

[17] LeVeque, R.J. Finite difference methods for ordinary and partial differential equations: steady state and time-dependent problems. SIAM, (2007)

[18] Mohanty, R.K., Sachdev, P.L., Jha, N. “An O (h4) accurate cubic spline TAGE method for non-linear singulat two point boundary value problems.” Appl. Math. Comput. 158, 853--868 (2004)

[19] Mohanty, R.K., Jain, M.K., Dhall, D. “High accuracy cubic spline approximation for two dimensional quasi-linear elliptic boundary value problems.” Appl. Math. Model. 37, 155--171 (2013)

[20] Polyanin, A.D., Handbook of linear partial differential equations for engineers and scientists, CRC press (2010)

[21] Polyanin, A.D., Kutepov, A.M., Vyazmin, A.V., Kazenin, D.A. Hydrodynamics, Mass and heat transfer in chemical engineering. Gordon & Breach Sci. Publ., London (2001)

[22] Polyanin, A.D., Zaitsev, V.F. Handbook of nonlinear partial differential equations, Chapman & Hall/CRC Press (2003)

[23] Rashidinia, J., Mohammadi ,R., Jalilian, R. “Spline methods for the solution of hyperbolic equation with variable coefficients.” Numer. Meth. Part. D. E. 23, 1411--1419 (2007)

[24] Saldanha, G., Ananthakrishnaiah, U. “A fourth-order finite difference scheme for two- dimensional nonlinear elliptic partial differential equations.” Numer. Meth. Part. D. E. 11, 33-- 40 (1995)

[25] Siraj-ul-Islam, Tirmizi, I.A. “A smooth approximation for the solution of special non-linear third-order boundary-value problems based on non-polynomial spline.” Int. J. Comput. Math. 83(4), 397--407 (2006)

[26] Varga R.S. Matrix Iterative Analysis. Springer Series in Computational Mathematics. Springer Berlin, Germany (2000)

[27] Weiser, A., Wheeler, M. F. “On convergence of block-centered finite differences for elliptic problems.” SIAM J. Numer. Anal. 25, 351--375 (1988)

[28] Young, D.L., Chang, T.J., Eldho, T.I. “The Riemann complex boundary element method for the solutions of two-dimensional Elliptic equations.” Appl. Math. Model. 26, 893--911 (2002)

[29] Young, D.M. Iterative solution of large linear systems. Courier Dover Pub., 2013.

[30] Zhao, J., Zhang, T., Corless, R.M. “Convergence of the compact finite difference method for second-order elliptic equations.” Appl. Math. Comput. 182, 1454--1469 (2006)