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Research Papers: Conduction

Comparative Assessment of the Finite Difference, Finite Element, and Finite Volume Methods for a Benchmark One-Dimensional Steady-State Heat Conduction Problem

[+] Author and Article Information
Sandip Mazumder

Fellow ASME
Department of Mechanical and
Aerospace Engineering,
The Ohio State University,
Suite E410, Scott Laboratory,
201 West 19th Avenue,
Columbus, OH 43210
e-mail: mazumder.2@osu.edu

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received October 5, 2016; final manuscript received December 26, 2016; published online March 15, 2017. Assoc. Editor: Alan McGaughey.

J. Heat Transfer 139(7), 071301 (Mar 15, 2017) (10 pages) Paper No: HT-16-1631; doi: 10.1115/1.4035713 History: Received October 05, 2016; Revised December 26, 2016

The finite difference (FD), finite element (FE), and finite volume (FV) methods are critically assessed by comparing the solutions produced by the three methods for a simple one-dimensional steady-state heat conduction problem with heat generation. Three issues are assessed: (1) accuracy of temperature, (2) accuracy of heat flux, and (3) satisfaction of global energy conservation. It is found that if the order of accuracy of the numerical discretization schemes is the same (central difference for FD and FV, linear basis functions for FE), the accuracy of the temperature produced by the three methods is similar, except close to the boundaries where the FV method outshines the other two methods. Consequently, the FV method is found to predict more accurate heat fluxes at the boundaries compared to the other two methods and is found to be the only method that guarantees both local and global conservation of energy irrespective of mesh size. The FD and FE methods both violate energy conservation, and the degree to which energy conservation is violated is found to be mesh size dependent. Furthermore, it is shown that in the case of prescribed heat flux (Neumann) and Newton cooling (Robin) boundary conditions, the accuracy of the FD method depends in large part on how the boundary condition is implemented. If the boundary condition and the governing equation are both satisfied at the boundary, the predicted temperatures are more accurate than in the case where only the boundary condition is satisfied.

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Figures

Grahic Jump Location
Fig. 1

Schematic representation of the geometry and boundary conditions used for the one-dimensional test case, along with the finite difference (or finite element) mesh. Both dimensional and nondimensional representations are shown.

Grahic Jump Location
Fig. 2

Schematic representation of the finite volume mesh in nondimensional coordinates showing cell-centers (squares) and cell faces (circles). The cell faces are at the same locations as the finite difference nodes.

Grahic Jump Location
Fig. 3

Error in nondimensional temperature incurred by the three methods for two different mesh sizes with prescribed temperature boundary conditions at both ends

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