Finite volume element method for analysis of unsteady reaction–diffusion problems

Abstract

A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the highgradient boundary layers.

Numerical Analysis of a Finite Volume-element Method for Unsteady Diffusion-reaction Equation

“…Finally, the trapezoidal rule is used to calculate the gradient quantities at cell faces. This paper can be considered as the continuation of the previous work [10], where the consistency, the stability, and the convergence of the finite volume-element method for solving the unsteady scalar reaction-diffusion equation are proved. The presentation of the paper starts from explanation of the theoretical formulation in Section 2.…”

“…The finite difference approximation and the divergence theorem are then applied to the temporal and spatial terms [10,11], respectively, of the above equation to yield (6) where is the unit outward normal vector of . The approximations to the cell average of i at time t n and t n+1 [12,13] i .…”

“…Recently, many researchers proposed some mixed techniques of the finite element and finite volume methods for analyzing the parabolic equation [8][9][10]. Ma et al [8] proposed the semi-discrete and full-discrete symmetric finite volume schemes based on the linear finite elements for a class of parabolic problems.…”

“…The finite volume schemes are also discretized on dual mesh and the nodal value at the center of control volume is approximated by linear finite element interpolation functions. Recently, Phongthanapanich and Dechaumphai [10] proposed a finite volume element method, that combines the finite volume and the finite element methods for analyzing the unsteady scalar reaction-diffusion equation. The cell-centered finite volume method is used to analyze the problem on a uniform control volume.…”

An explicit finite volume element method for solving characteristic level set equation on triangular grids

“…Detailed explanation of the procedure for approximating this gradient term is described in Ref. [20] where it was applied to the unsteady reaction-diffusion problems.…”