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A novel numerical method for the solution of nonlinear equations with applications to heat transfer

J.N. Reddy (J. Mike Walker Department of Mechanical Engineering, Texas A&M University, College Station, Texas, USA)
Matthew Martinez (J. Mike Walker Department of Mechanical Engineering, Texas A&M University, College Station, Texas, USA)
Praneeth Nampally (J. Mike Walker Department of Mechanical Engineering, Texas A&M University, College Station, Texas, USA)

International Journal of Numerical Methods for Heat & Fluid Flow

ISSN: 0961-5539

Article publication date: 11 February 2021

Issue publication date: 24 May 2021

225

Abstract

Purpose

The purpose of this study is to extend a novel numerical method proposed by the first author, known as the dual mesh control domain method (DMCDM), for the solution of linear differential equations to the solution of nonlinear heat transfer and like problems in one and two dimensions.

Design/methodology/approach

In the DMCDM, a mesh of finite elements is used for the approximation of the variables and another mesh of control domains for the satisfaction of the governing equation. Both meshes fully cover the domain but the nodes of the finite element mesh are inside the mesh of control domains. The salient feature of the DMCDM is that the concept of duality (i.e. cause and effect) is used to impose boundary conditions. The method possesses some desirable attributes of the finite element method (FEM) and the finite volume method (FVM).

Findings

Numerical results show that he DMCDM is more accurate than the FVM for the same meshes used. Also, the DMCDM does not require the use of any ad hoc approaches that are routinely used in the FVM.

Originality/value

To the best of the authors’ knowledge, the idea presented in this work is original and novel that exploits the best features of the best competing methods (FEM and FVM). The concept of duality is used to apply gradient and mixed boundary conditions that FVM and its variant do not.

Keywords

Acknowledgements

The research reported herein was supported by the O’Donnell Foundation Chair IV held by the first author.

Data availability statement. The data that support the findings of this study are available from the corresponding author upon reasonable request.

Citation

Reddy, J.N., Martinez, M. and Nampally, P. (2021), "A novel numerical method for the solution of nonlinear equations with applications to heat transfer", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 31 No. 6, pp. 1884-1904. https://doi.org/10.1108/HFF-07-2020-0397

Publisher

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Emerald Publishing Limited

Copyright © 2021, Emerald Publishing Limited

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