A higher-order orthogonal collocation technique for discontinuous two-dimensional problems with Neumann boundary conditions

Bhal, Santosh Kumar; Nandi, Ashish Kumar; Rababah, Abedallah

doi:10.22034/cmde.2024.60344.2577

A higher-order orthogonal collocation technique for discontinuous two-dimensional problems with Neumann boundary conditions

Document Type : Research Paper

Authors

¹ Department of Mathematics, School of Advanced Sciences and Languages, Vellore Institute of Technology, Bhopal, India.

² Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Chennai, Tamil Nadu-600127, India.

³ Department of Mathematical Sciences, United Arab Emirates University, United Arab Emirates.

10.22034/cmde.2024.60344.2577

Abstract

In this paper, the orthogonal spline collocation method (OSCM) is employed to address the solution of the Helmholtz equation in two-dimensional problems. It is characterized by discontinuous coefficients with certain wave numbers. The solution is approximated by employing distinct basis functions, namely, monomial along the x-direction and Hermite along the y-direction. Additionally, to solve the two-dimensional problems efficiently in the sense of computational cost with fewer operation counts, the matrix decomposition algorithm (MDA) is used to convert them into a set of one-dimensional problems. As a consequence, the resulting reduced matrix becomes non-singular in discrete cases. To assess the performance of the proposed numerical scheme, a grid refinement analysis is conducted to incorporate various wave coefficients of the Helmholtz equation. The illustrations and examples demonstrate a higher order of convergence compared to existing methods.

Keywords

Main Subjects

Numerical methods for partial differential equations, boundary value problems

References

[1] A. Aghili and A. H. Shabani, Application of the orthogonal collocation method in evaluating the rate of reactions using thermogravimetric analysis data Thermochimica Acta, (2024), 179698.
[2] S. K. Bhal and P. Danumjaya, A Fourth-order orthogonal spline collocation solution to 1D-Helmholtz equation with discontinuity, The Journal of Analysis, 27 (2019), 377–390.
[3] S. K. Bhal, P. Danumjaya, and G. Fairweather, A fourth–order orthogonal spline collocation method for twodimensional Helmholtz problems with interfaces, Numerical Methods for Partial Differential Equations, 36(6) (2020), 1811–1829.
[4] S. K. Bhal, P. Danumjaya, and G. Fairweather, High-order orthogonal spline collocation methods for two-point boundary value problems with interfaces, Mathematics and Computers in Simulation, 174 (2020), 102–122.
[5] S. K. Bhal, P. Danumjaya, and A. Kumar, A fourth-order orthogonal spline collocation method to fourth-order boundary value problems, International Journal for Computational Methods in Engineering Science and Mechanics, 20(5) (2019), 460–470.
[6] B. Bialecki and G. Fairweather, Matrix decomposition algorithms for separable elliptic boundary value problems in two space dimensions, Journal of Computational and Applied Mathematics, 46(3) (1993), 369–386.
[7] B. Bialecki and G. Fairweather, Matrix decomposition algorithms in orthogonal spline collocation for separable elliptic boundary value problems, SIAM Journal on Scientific Computing, 16(2) (1995), 330–347.
[8] B. Bialecki and G. Fairweather, Orthogonal spline collocation methods for partial differential equations, Journal of Computational and Applied Mathematics, 128 (2001), 55–82.
[9] B. Bialecki, G. Fairweather, and K. R. Bennett, Fast direct solvers for piecewise Hermite bicubic orthogonal spline collocations equations, SIAM Journal on Numerical Analysis, 29(1) (1992), 156–173.
[10] B. Bialecki, G. Fairweather, and A. Karageorghis, Matrix decomposition algorithms for elliptic boundary value problems: a survey, Numerical Algorithms, 56 (2011), 253–295.
[11] B. Bialecki, G. Fairweather, and K. A. Remington, Fourier methods for piecewise Hermite bicubic orthogonal spline collocation, East-West Journal of Numerical Mathematics, 2 (1994), 1–20.
[12] R. Dautray and J. L. Lions, Mathematical analysis and numerical methods for science and technology, SpringerVerlag, New York, 1990.
[13] C. De Boor and B. Swartz, Collocation at Gauss points, SIAM Journal on Numerical Analysis, 10(4) (1973), 582–606.
[14] J. C. Diaz, G. Fairweather, and P. Keast, Algorithm 603 COLROW and ARCECO: Packages for solving certain almost block diagonal linear systems by modified alternate row and column elimination, ACM Transactions on Mathematical Software, 9(3) (1983), 376–380.
[15] D. S. Dillery, High order orthogonal spline collocation schemes for elliptic and parabolic problems, Thesis (Ph.D.), University of Kentucky, 1994.
[16] W. R. Dyksen, Tensor product generalized ADI methods for separable elliptic problems, SIAM Journal on Numerical Analysis, 24(1) (1987), 59–76.
[17] G. Fairweather and D. Meade, A survey of spline collocation methods for the numerical solution of differential equations, Mathematics for Large Scale Computing, CRC press, (1989), 297–341.
[18] X. Feng, A high-order compact scheme for the one-dimensional Helmholtz equation with a discontinuous coefficient, International Journal of Computer Mathematics, 89(5) (2012), 618–624.
[19] L. Feng, F. Geng, and L. Yao, Numerical solutions of Schr¨odinger-Boussinesq system by orthogonal spline collocation method, Journal of Computational and Applied Mathematics, (2024), 115984.
[20] X. Feng, Z. Li, and Z. Qiao, High-order compact finite difference schemes for the Helmholtz equations with discontinuous coefficients, Journal of Computational Mathematics, 29 (2011), 324–340.
[21] Y. Fu, Compact fourth-order finite difference schemes for Helmholtz equation with high wave numbers, Journal of Computational Mathematics, (2008), 98–111.
[22] B. Guy, G. Fibich, and S. Tsynkov, High-order numerical method for the nonlinear Helmholtz equation with material discontinuities in one space dimension, Journal of Computational Physics, 227(1) (2007), 820–850.
[23] B. Guy, G. Fibich, S. Tsynkov, and E. Turkel, Fourth order schemes for time-harmonic wave equations with discontinuous coefficients, Communications in Computational Physics, 5 (2009), 442–455.
[24] F. Ihlenburg and I. Babuˇska, Finite element solution of the Helmholtz equation with high wave number Part I: The h-version of the FEM, Computers & Mathematics with Applications, 30(9) (1995), 9–37.
[25] K. Ito, Z. Qiao, and J. Toivanen, A domain decomposition solver for acoustic scattering by elastic objects in layered media, Journal of Computational Physics, 227(19) (2008), 8658–8698.
[26] M. C. Junger and D. Feit, Sound, structures and their interaction, Cambridge, MA: MIT press, 1986.
[27] P. M. Prenter and R. D. Russell, Orthogonal collocation for elliptic partial differential equations, SIAM Journal on Numerical Analysis, 13(6) (1976), 923–939.
[28] A. Rakhim, Existence and uniqueness for a two-point interface boundary value problem, Electronic Journal of Differential Equations, 242 (2013), 1–12.
[29] M. P. Robinson and G. Fairweather, Orthogonal cubic spline collocation solution of underwater acoustic wave propagation problems, Journal of Computational Acoustics, 1(3) (1993), 355–370.
[30] E. Roubine, Étude des ondes electromagnetiques guidees par les circuits en helice, Annales des Télécommunications, 7 (1952), 206–216.
[31] I. H. Sloan, D. Tran, and G. Fairweather, A fourth-order cubic spline method for linear second-order two point boundary value problems, IMA Journal of Numerical Analysis, 13(4) (1993), 591–607.
[32] A. Uri, S. Pruess, and R. D. Russell, On spline basis selection for solving differential equations, SIAM Journal on Numerical Analysis, 20(1) (1983), 121–142.
[33] R. Wang, Y. Yan, A. S. Hendy, and L. Qiao, BDF2 ADI orthogonal spline collocation method for the fractional integro-differential equations of parabolic type in three dimensions, Computers & Mathematics with Applications, 155 (2024), 126–141.