# A Comparative Study of Characteristic Finite Element and Characteristic Finite Volume Methods for Convection-Diffusion-Reaction Problems on Triangular Grids

## Main Article Content

## Abstract

The paper aims to compare the accuracy and robustness of the characteristic finite element method (CFEM) and characteristic finite volume method (CFVM) for solving convection-diffusion-reaction problems on two-dimensional triangular grids. The tests are performed on a square unit domain, to which an advective field is imposed in a domain. The results show that the CFEM gives less accurate solution than CFVM for the rotation of a slotted-cylinder and rotation of Gaussian cone problems. Moreover, CFEM gives oscillate solution while the CFVM provides an oscillation-free solution for the skew flow to the grid problem.

## Article Details

How to Cite

*Applied Science and Engineering Progress*,

*12*(4), 235–242. Retrieved from https://ph02.tci-thaijo.org/index.php/ijast/article/view/232518

Issue

Section

Research Articles

## References

[1] A. N. Brooks and T. J. R. Hughes, “Streamline upwind Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equation,” Computer Methods in Applied Mechanics and Engineering, vol. 32, pp. 199–259, 1982.

[2] R. Codina, “A discontinuity-capturing crosswind dissipation for the finite element solution of the convection-diffusion equation,” Computer Methods in Applied Mechanics and Engineering, vol. 110, pp. 325–342, 1993.

[3] O. C. Zienkiewicz and R. Codina, “A general algorithm for compressible and incompressible flow - part I. the split, characteristic based scheme,” International Journal for Numerical Methods in Fluids, vol. 20, pp. 869–885, 1995.

[4] R. Codina, “Comparison of some finite element methods for solving the diffusion-convectionreaction equation,” Computer Methods in Applied Mechanics and Engineering, vol.156, no. 1–4, pp. 185–210, 1998.

[5] S. Phongthanapanich and P. Dechaumphai, “A characteristic-based finite volume element method for convection-diffusion-reaction equation,” Transactions of the Canadian Society for Mechanical Engineering, vol. 32, pp. 549–560, 2008.

[6] S. Phongthanapanich and P. Dechaumphai, “Explicit characteristic finite volume method for convectiondiffusion equation on rectangular grids,” Journal of the Chinese Institute of Engineers, vol. 34, pp. 239–252, 2011.

[7] S. Phongthanapanich and P. Dechaumphai, “An explicit characteristic finite volume element method for non-divergence free convectiondiffusion- reaction equation,” International Journal of Advances in Engineering Sciences and Applied Mathematics, vol. 4, pp. 179–192, 2012.

[8] Finite Volume Methods, Elsevier Science, North Holland, Netherlands, 2000, pp. 715–1022.

[9] J. Chessa and T. Belytschko, “An extended finite element method for two-phase fluids,” Transaction of ASME, vol. 70, pp. 10–17, 2003.

[10] S. T. Zalesak, “Fully multidimensional flux-corrected transport algorithms for fluids,” Journal of Computational Physics, vol. 31, pp. 335–362, 1979.

[11] H. Wang and J. Liu, “Development of CFL-free, explicit schemes for multidimensional advectionreaction equations,” SIAM Journal on Scientific Computing, vol. 23, pp.1418–1438, 2001.

[12] G. Hauke, “A simple subgrid scale stabilized method for the advection-diffusion-reaction equation,” Computer Methods in Applied Mechanics and Engineering, vol. 191, pp. 2925–2947, 2002.

[2] R. Codina, “A discontinuity-capturing crosswind dissipation for the finite element solution of the convection-diffusion equation,” Computer Methods in Applied Mechanics and Engineering, vol. 110, pp. 325–342, 1993.

[3] O. C. Zienkiewicz and R. Codina, “A general algorithm for compressible and incompressible flow - part I. the split, characteristic based scheme,” International Journal for Numerical Methods in Fluids, vol. 20, pp. 869–885, 1995.

[4] R. Codina, “Comparison of some finite element methods for solving the diffusion-convectionreaction equation,” Computer Methods in Applied Mechanics and Engineering, vol.156, no. 1–4, pp. 185–210, 1998.

[5] S. Phongthanapanich and P. Dechaumphai, “A characteristic-based finite volume element method for convection-diffusion-reaction equation,” Transactions of the Canadian Society for Mechanical Engineering, vol. 32, pp. 549–560, 2008.

[6] S. Phongthanapanich and P. Dechaumphai, “Explicit characteristic finite volume method for convectiondiffusion equation on rectangular grids,” Journal of the Chinese Institute of Engineers, vol. 34, pp. 239–252, 2011.

[7] S. Phongthanapanich and P. Dechaumphai, “An explicit characteristic finite volume element method for non-divergence free convectiondiffusion- reaction equation,” International Journal of Advances in Engineering Sciences and Applied Mathematics, vol. 4, pp. 179–192, 2012.

[8] Finite Volume Methods, Elsevier Science, North Holland, Netherlands, 2000, pp. 715–1022.

[9] J. Chessa and T. Belytschko, “An extended finite element method for two-phase fluids,” Transaction of ASME, vol. 70, pp. 10–17, 2003.

[10] S. T. Zalesak, “Fully multidimensional flux-corrected transport algorithms for fluids,” Journal of Computational Physics, vol. 31, pp. 335–362, 1979.

[11] H. Wang and J. Liu, “Development of CFL-free, explicit schemes for multidimensional advectionreaction equations,” SIAM Journal on Scientific Computing, vol. 23, pp.1418–1438, 2001.

[12] G. Hauke, “A simple subgrid scale stabilized method for the advection-diffusion-reaction equation,” Computer Methods in Applied Mechanics and Engineering, vol. 191, pp. 2925–2947, 2002.