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A New High Accuracy Off-Step Discretisation for the Solution of 2D Nonlinear Triharmonic Equations

Published online by Cambridge University Press:  28 May 2015

Swarn Singh ,
Suruchi Singh  and
R. K. Mohanty
Swarn Singh
Affiliation:
Department of Mathematics, Sri Venkateswara College, University of Delhi, New Delhi-110021, India
Suruchi Singh
Affiliation:
Department of Mathematics, Aditi Mahavidayalaya, University of Delhi, Delhi-110039, India
R. K. Mohanty*
Affiliation:
Department of Applied Mathematics, South Asian University, Akbar Bhawan, Chanakyapuri, New Delhi - 110021, India
*
Corresponding author. Email Address: rmohanty@sau.ac.in
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Abstract

In this article, we derive a new fourth-order finite difference formula based on off-step discretisation for the solution of two-dimensional nonlinear triharmonic partial differential equations on a 9-point compact stencil, where the values of u,(∂2u/∂n2) and (∂4u/∂n4) are prescribed on the boundary. We introduce new ways to handle the boundary conditions, so there is no need to discretise the boundary conditions involving the partial derivatives. The Laplacian and biharmonic of the solution are obtained as a by-product of our approach, and we only need to solve a system of three equations. The new method is directly applicable to singular problems, and we do not require any fictitious points for computation. We compare its advantages and implementation with existing basic iterative methods, and numerical examples are considered to verify its fourth-order convergence rate.

Keywords

65N06High accuracy finite differencesoff-step discretisationtwo-dimensional nonlinear triharmonic equationsLaplacianbiharmonictriharmonicmaximum absolute errors
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 3 , Issue 3 , August 2013 , pp. 228 - 245
Copyright
Copyright © Global-Science Press 2013

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References

[1]Bauer, L. and Riess, E.L., Block five diagonal matrices and the fast numerical solution of the biharmonic equation, Math. Comp. 26, 311–326 (1972).Google Scholar
[2]Ehrlich, L.W., Solving the biharmonic equation as coupled finite difference equations, SIAM. J. Num. Anal. 8, 278–287 (1971).Google Scholar
[3]Ehrlich, L.W., Point and block SOR applied to a coupled set of difference equations, Computing 12, 181–194 (1974).Google Scholar
[4]Evans, D.J. and Mohanty, R.K., Block iterative methods for the numerical solution of two-dimensional nonlinear biharmonic equations, Int. J. Comput. Math. 69, 371–390 (1998).Google Scholar
[5]Hageman, L.A. and Young, D.M., Applied Iterative Methods, Academic Press (1981).Google Scholar
[6]Kelly, C.T., Iterative Methods for Linear and Non-linear Equations, SIAM publications, Philadelphia (1995).Google Scholar
[7]Kwon, Y., Manohar, R. and Stephenson, J.W., Single cell fourth order methods for the biharmonic equation, Congress Numerantium 34, 475–482 (1982).Google Scholar
[8]Meurant, G., Computer Solution of Large Linear Systems, North-Holland (1999).Google Scholar
[9]Mohanty, R.K., Jain, M.K. and Pandey, P.K., Finite difference methods of order two and four for 2D nonlinear biharmonic problems of first kind, Int. J. Comput. Math. 61, 155–163 (1996).Google Scholar
[10]Mohanty, R.K. and Pandey, P.K., Difference methods of order two and four for systems of mildly nonlinear biharmonic problems of second kind in two space dimensions, Numer. Meth. Partial Diff. Eq. 12, 707–717 (1996).Google Scholar
[11]Mohanty, R.K. and Pandey, P.K., Families of accurate discretisation of order two anfour for 3D mildlyy nonlinear biharmonic problems of second kind, Int. J. Comput. Math. 68, 363–380 (1998).CrossRefGoogle Scholar
[12]Mohanty, R.K., Evans, D.J. and Pandey, P.K., Block iterative methods for the numerical solution of three-dimensional nonlinear biharmonic equations of first kind, Int. J. Comput. Math. 77, 319–332 (2001).Google Scholar
[13]Mohanty, R.K. and Singh, S., A new fourth order discretisation for singularly perturbed two-dimensional nonlinear elliptic boundary value problems, Applied Math. Comp. 175, 1400–1414 (2006).Google Scholar
[14]Mohanty, R.K. and Singh, S., A new highly accurate discretisation for three dimensional singularly perturbed nonlinear elliptic partial differential equation, Numer. Meth. Partial Diff. Eq. 22, 1379–1395 (2006).Google Scholar
[15]Mohanty, R.K., A new high accuracy finite difference discretisation for the solution of 2D nonlinear biharmonic equations using coupled approach, Numer. Meth. Partial Diff. Eq. 26, 931–944 (2010).Google Scholar
[16]Mohanty, R.K., Single-cell compact finite-difference discretisation of order two and four for multidimensional triharmonic problems, Numer. Meth. Partial Diff. Eq. 26, 1420–1426 (2010).Google Scholar
[17]Singh, S., Khattar, D. and Mohanty, R.K., A new coupled approach high accuracy method for the solution of 2D nonlinear biharmonic equations, Neural Parallel and Scientific Computation 17, 239–256 (2009).Google Scholar
[18]Khattar, D., Singh, S. and Mohanty, R.K., A new coupled approach high accuracy method for the solution of 3D nonlinear biharmonic equations, Applied Math. Comp. 215, 3036–3044 (2009).Google Scholar
[19]Mohanty, R.K., Jain, M.K. and Mishra, B.N., A compact discretisation of O(h4) for two-dimensional nonlinear triharmonic equations, Physica Scripta 84, ID. 025002 (2011).Google Scholar
[20]Mohanty, R.K., Jain, M.K. and Mishra, B.N., A novel method of O(h4) for three-dimensional nonlinear triharmonic equations, Commun. Comput. Phys. 12, 1417–1433 (2012).Google Scholar
[21]Jain, M.K., Numerical Solution of Differential Equations, 2nd Ed., Wiley Eastern Limited, New Delhi (1984).Google Scholar
[22]Collatz, L., The Numerical Traetment of Differential Equations, 3rd Ed., Springer Verlag (1966).Google Scholar
[23]Ames, W.F., Numerical Methods for Partial Differential Equations, 2nd Ed., Academic Press (1977).Google Scholar
[24]Parter, S.V., Block Iterative Methods in Elliptic Problem Solvers, Schultz, M.H. Ed., Academic Press (1981).Google Scholar
[25]Saad, Y., Iterative Methods for Sparse Linear Systems, PWS publishing company (1996).Google Scholar
[26]Smith, J., The coupled equation approach to the numerical solution of the biharmonic equation by finite differences, SIAM. J. Num. Anal. 5, 104–111 (1970).CrossRefGoogle Scholar
[27]Spotz, W.F. and Carey, G.F., High-Order compact scheme for the steady stream-function vorticity equations, Int. J. Numer. Meth. Engg. 38, 3497–3512 (1995).Google Scholar
[28]Stephenson, J.W., Single cell discretisation of order two and four for biharmonic problems, J. Comput. Phys. 55, 65–80 (1984).CrossRefGoogle Scholar
[29]Thomas, J.W., Numerical Partial Differential Equations: Conservation Laws and Elliptic Equations, Springer Verlag (1999).Google Scholar
[30]Strikwerda, J.C., Finite Difference Schemes and Partial Differential Equations, SIAM (2004).Google Scholar
[31]Varga, R.S., Matrix Iterative Analysis, Springer Verlag (2000).CrossRefGoogle Scholar