A fourth order finite difference method for a nonlinear Helmholtz type boundary value problems in PDEs |
|
| Author |
pramod_10p@hotmail.com, oman1482@hotmail.com
Department of Mathematics, Dyal Singh College, University of Delhi, India; Ministry of Higher Education, Sultanate of Oman
|
| Abstract |
In this paper we present a fourth order finite difference method for solving nonlinear
Helmholtz type elliptic boundary value problems in two dimensions subject to the Dirichlet
boundary conditions. The present fourth order method is based on the approximation of
derivative by finite difference method and Helmholtz equation. The truncation error and
convergence analysis are presented for the proposed method. We present numerical experiments
to demonstrate the efficiency and accuracy of the method.
|
| Keywords |
finite difference method; fourth order method; Helmholtz equations; maximum absolute error;
nonlinear elliptic equation
|
| DOI |
doi:10.15330/ms.45.2.213-224
|
| Reference |
1. Britt D.S., Tsynkov S.V., Turkel E. A high-order numerical method for the Helmholtz equation with
nonstandard boundary conditions// SIAM J. Sci. Comput. – 2013. – V.35, ¹5. – P. 2255–2292.
2. Ernst O.G., Gander M.J. Why it is difficult to solve Helmholtz problems with classical iterative methods// https://www.tu-chemnitz.de/mathematik/numa/PubArchive/helmholtzDurham.pdf 3. Medvinsky M., Tsynkov S., Turkel E. The method of difference potentials for the Helmholtz equation using compact high order schemes// J. Sci. Comput. – 2012. – V.53. – P. 150–193. 4. Singer I., Turkel E. High-order finite difference methods for the Helmholtz equation// Computer Methods in Applied Mechanics and Engineering. – 1998. – V.163, ¹1-4. – P. 343–358. 5. Nabavi M., Siddiqui M.H.K., Dargahi J. A new 9-point sixth-order accurate compact finite-difference method for the Helmholtz equation// Journal of Sound and Vibration. – 2007. – V.307, ¹3–5. – P. 972– 982. 6. Godehard S. Compact finite difference schemes of sixth order for the Helmholtz equation// Journal of Computational and Applied Mathematics. – V.203, ¹1. – P. 15–31. 7. Singer I., Turkel E. Sixth-order accurate finite difference schemes for the Helmholtz equation// J. Comp. Acous. – 2006. – V.14. – P. 339–352. 8. Collatz L., Numerical treatment of differential equations (3/e), Springer Verlag, Berlin, 1966. 9. Pandey P.K. a rational finite difference approximation of high order accuracy for nonlinear two point boundary value problems// Sains Malaysiana. – 2014. – V.43, ¹7. – P. 1105–1108. 10. Jain M.K., Jain R.K., Mohanty R.K. Fourth order difference methods for the system of 2-D nonlinear elliptic partial differential equations// Numer. Methods Partial Differential Equations. – 1991. – V.7. – P. 227–244. 11. Jain M.K., Iyenger S.R.K., Jain R.K., Numerical methods for scientific and engineering computations (2/e), Wiley Eastern Ltd., New Delhi, 1987. 12. Henrici P., Discrete variables methods in ordinary differential equation, John Wiley and Sons, New York, 1962. 13. Varga, R.S., Matrix Iterative Analysis, Second Revised and Expanded Edition, Springer-Verlag, Heidelberg, 2000. 14. Volkov, Y.S., Miroshnichenko, V.L. Norm estimates for the inverses of matrices of monotone type and totally positive matrics// Siberian Mathematical Journal. – 2009. – V.50, ¹6. – P. 982–987. 15. Ahlberg J.H., Nilson E.N. Convergence properties of the spline fit// J. SIAM. – 1963. – V.11, ¹1. – P. 95–104. |
| Pages |
213-224
|
| Volume |
45
|
| Issue |
2
|
| Year |
2016
|
| Journal |
Matematychni Studii
|
| Full text of paper | |
| Table of content of issue |