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.

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 less operation counts,
matrix decomposition algorithm (MDA) is used to convert it 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

Computational Methods for Differential Equations

Articles in Press, Accepted Manuscript
Available Online from 23 September 2024

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History

  • Receive Date: 30 January 2024
  • Revise Date: 17 August 2024
  • Accept Date: 31 August 2024

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