Effect of boundary condition approximation on convergence and accuracy of a finite volume discretization of the transient heat conduction equation

Abstract

Purpose

The purpose of this paper is to investigate the effect of numerical boundary condition implementation on local error and convergence in L2-norm of a finite volume discretization of the transient heat conduction equation subject to several boundary conditions, and for cases with volumetric heat generation, using both fully implicit and Crank-Nicolson time discretizations. The goal is to determine which combination of numerical boundary condition implementation and time discretization produces the most accurate solutions with the least computational effort.

Design/methodology/approach

The paper studies several benchmark cases including constant temperature, convective heating, constant heat flux, time-varying heat flux, and volumetric heating, and compares the convergence rates and local to analytical or semi-analytical solutions.

Findings

The Crank-Nicolson method coupled with second-order expression for the boundary derivatives produces the most accurate solutions on the coarsest meshes with the least computation times. The Crank-Nicolson method allows up to 16X larger time step for similar accuracy, with nearly negligible additional computational effort compared with the implicit method.

Practical implications

The findings can be used by researchers writing similar codes for quantitative guidance concerning the effect of various numerical boundary condition approximations for a large class of boundary condition types for two common time discretization methods.

Originality/value

The paper provides a comprehensive study of accuracy and convergence of the finite volume discretization for a wide range of benchmark cases and common time discretization methods.

Citation

Guillot, M.J. and McCool, S.C. (2015), "Effect of boundary condition approximation on convergence and accuracy of a finite volume discretization of the transient heat conduction equation", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 25 No. 4, pp. 950-972. https://doi.org/10.1108/HFF-02-2014-0033