Cubic spline Numerov type approach for solution of Helmholtz equation

Subject Areas : Difference and functional equations

J. Rashidinia ¹ (Department of Mathematics,College of basic science, Islamic Azad University, Alborz, Iran)
H. S. Shekarabi ² (Department of Mathematics,College of basic science, Islamic Azad University, Alborz, Iran)
M. Aghamohamadi ³ (Department of Mathematics,College of basic science, Islamic Azad University, Alborz, Iran)

Received: 2014-05-14 Accepted : 2014-08-28 Published : 2014-03-01

Keywords: Cubic spline, Numerov type, Stability, Helmholtz equation, Finite difference,

Abstract :

We have developed a three level implicit method for solution of the Helmholtz equation. Using the cubic spline in space and finite difference in time directions. The approach has been modified to drive Numerov type finite difference method. The method yield the tri-diagonal linear system of algebraic equations which can be solved by using a tri-diagonal solver. Stability and error estimation of the presented method are analyzed. The obtained results satised the ability and effciency of the method.

References:

[1] Carlos J. S. Alves, Svilen S. Valtchev,Numerical simulation of acoustic wave scattering using a meshfree plane waves method,International Workshop on MeshFree Methods( 2003),1-6.

 

[2] K. Atkinson, W. Han, Theoretical Numerical Analysis: A Functional Analysis Framework, Springer,(2005).

 

[3] R. J. Astley, P. Gamallo, Special short elements for flow acoustics, Comput. Method Appl. Mech. Engrg. 194 (2005), 341-353.

[4] R. K. Beatson, J. B. Cherrie, C. T. Mouat, Fast fitting of radial basis functions: method based on preconditioned GMRES iteration, Adv. Comput. Math. 11 (1999), 253-270.

 

[5] R. K. Beatson, W. A. Light, S. Billings, Fast solution of the radial basis function interpolation equations: domain decomposition methods, SIAM J. Sci. Comput. 5 (2000),1717-1740.

 

[6] A. I. Bouhamid, A. Le Mhaut, Spline curves and surfaces under tension, (1994),51-58.

 

[7] A. I. Bouhamid, A. Le Mhaut, Multivariate interpolating (m;s)-spline, Adv. Comput. Math. 11 (1999), 287-314.

 

[8] G. M. L. Gladwell, N. B. Willms, On the mode shape of the Helmholtz equation, J. Sound Vib. 188(1995), 419-433.

 

[9] Charles I. Goldstein, A Finite Element Method for Solving Helmholtz,Type Equationsin Waveguides and Other Unbounded Domains, mathematics of computation,39 (160) (1982), 309-324. 

 

[10] F. Ihlenburg, I. Babusk. Finite element solution of the Helmholtz equation with high wave number part I: the hversion of the FEM. Computers Mathematics with Applications, 30 (9) (1995), 9-37.

 

[11] F. Ihlenburg, I. Babuska. Finite element solution of the Helmholtz equation with high wave number part II: the hp version of the FEM. SIAM Journal of Numerical Analysis, 34 (1) (1997), 315-358.

 

[12] M. K. Jain, Numerical Solution of Dierential Equations, 2nd edn. Wiley, New Delhi (1984).

 

[13] E. J. Kansa, A scattered data approximation scheme with applications to computational fluid dynamics. I. Surface approximations and partial derivative estimates, Comput. Math. Appl. 19 (8,9) (1990),127-145.

 

[14] E. J. Kansa, Multiquadrics a scattered data approximation scheme with applications to computational fluid dynamics. II. Solutions to parabolic, hyperbolic partial differential equations, Comput. Math. Appl. 19 (8,9) (1990), 127-145.

 

[15] Y. C. Hon,C. S. Chen, Numerical comparisons of two meshless methods using radial basis functions engineering analysis with boundary elements. 26 (2002), 205-225.

 

[16] R. K. Mohanty, Stability interval for explicit difference schemes for multi-dimensional second order hyperbolic equations with significant first order space derivative terms, Appl. Math. Comput. 190 (2007),1683-1690.

 

[17] R. K. Mohanty, Venu Gopal, High accuracy cubic spline finite difference approximation for the solution of one-space dimensional non-linear wave equations,Applied Mathematics and Computation 218 (2011), 4234-4244.

 

[18] C. C. Paige, M. A. Saunders, LSQR: an algorithm for sparse linear equations and sparse least squares, ACM Trans. Math. Softw. 8 (1982) ,43-71.

 

[19] J. Rashidinia, R. Jalilian, V. Kazemi, Spline methods for the solutions of hyperbolic equations, Appl. Math. Comput. 190 (2007), 882-886.

 

[20] A. S. Wood, G. E. Tupholme, M. I. H. Bhatti, P. J. Heggs, Steady-state heattransfer through extended plane surfaces, Int. Commun. Heat Mass Transfer 22 (1995), 99-109.