A Robust Quintic Hermite Collocation Method for One-Dimensional Heat Conduction Equation

Year 2024, , 82 - 89, 31.08.2024

Selçuk Kutluay , Murat Yağmurlu , Ali Sercan Karakaş

https://doi.org/10.33187/jmsm.1475294

Abstract

In this work, a new robust numerical solution scheme constructed on Quintic Hermite Collocation Method (QHCM) utilizing the traditional Crank-Nicolson type approximation technique is developed for solving 1D heat conduction equation with certain initial and boundary conditions which is mostly handled as a prototype equation to support the reliability of many proposed new numerical methods. All temporal and spatial quantities in the equation are fully discretized using a usual Crank-Nicolson type finite difference approximation and a QHCM, respectively. In obtaining the present scheme, all the roots of the fourth degree Legendre and Chebyshev polynomials shifted to the unit interval are used as suitable inner collocation points. The obtained results from the developed scheme are found to be good enough and better than those from other schemes encountered in the literature. The scheme is also shown to be unconditionally stable by Fourier stability test.

Keywords

Collocation finite element method, Fourier stability test, Heat conduction equation, Legendre and Chebyschev roots, Quintic hermite spline basis functions

Ethical Statement

The authors of the present manuscript clearly declare that all of the methods and schemes used in the manuscript do not need any ethical committee and/or legal special requirement or permission.

Supporting Institution

This work has been supported by İnönü University Scientific Research Projects Unit under Grant No: FDK-2023-3402.

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