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Solving convection-diffusion-reaction equation by adaptive finite volume element method. (English) Zbl 1241.65073

Summary: A finite volume element method is combined with an adaptive meshing technique to solve the two-dimensional unsteady convection-diffusion-reaction equation. The finite volume method is used to derive the discretized equations while concept of the finite element technique is applied to determine the gradient quantities at cell faces. Second-order accuracy in both space and time are achieved by applying the Taylor’s series expansion along the local characteristic lines. An adaptive meshing technique is applied to further improve the solution accuracy, and to minimize the computational time and computer memory requirement. The efficiency of the adaptive finite volume element method is evaluated by the examples of pure-convection, convection-diffusion, convection-reaction, and diffusion-reaction problems.

MSC:

65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
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[1] Barth, T.; Ohlberger, M., Encyclopedia of Computational Mechanics Volume 1: Fundamentals (2004), John Wiley and Sons
[2] Carmo, E. G.D.; Alvarez, G. B., A new stabilized finite element formulation for scalar convection-diffusion problems: the streamline and approximate upwind/Petrov-Galerkin method, Comput. Meth. Appl. Mech. Eng., 192, 3379-3396 (2003) · Zbl 1054.76055
[3] Carmo, E. G.D.; Alvarez, G. B.; Rochinha, F. A.; Loula, A. F.D., Galerkin projected residual method applied to diffusion-reaction problems, Comput. Meth. Appl. Mech. Eng., 197, 4559-4570 (2008) · Zbl 1194.76113
[4] Codina, R., Comparison of some finite element methods for solving the diffusion-convection-reaction equation, Comput. Meth. Appl. Mech. Eng., 156, 185-210 (1998) · Zbl 0959.76040
[5] Dechaumphai, P., Adaptive finite element technique for heat transfer problems, J. Energ. Heat Mass Transfer, 17, 87-94 (1995)
[6] Dechaumphai, P., Finite Element Method: Fundamentals and Applications (2010), Alpha Science Intl. Ltd · Zbl 1213.65004
[7] Ilinca, F.; Hetu, J. F., A new stabilized finite element method for reaction-diffusion problems: the source-stabilized Petrov-Galerkin method, Int. J. Num. Meth. Eng., 75, 1607-1630 (2008) · Zbl 1158.76351
[8] Jasak, H.; Weller, H. G.; Gosman, A. D., High resolution NVD differencing scheme for arbitrarily unstructured meshes, Int. J. Num. Meth. Fluids, 31, 431-449 (1999) · Zbl 0952.76057
[9] Kruganov, A.; Tadmor, E., New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., 160, 241-282 (2000) · Zbl 0987.65085
[10] Lamine, S.; Edwards, M. G., High-resolution convection schemes for flow in porous media on highly distorted unstructured grids, Int. J. Num. Meth. Eng., 76, 1139-1158 (2008) · Zbl 1195.76266
[11] Leveque, R. J., Finite Volume Methods for Hyperbolic Problems (2005), Cambridge University Press
[12] Limtrakarn, W.; Dechaumphai, P., High-speed compressible flow analysis by adaptive cell-centered finite elements, J. Energ. Heat Mass Transfer, 24, 141-162 (2002)
[13] Malatip, A.; Wansophark, N.; Dechaumphai, P., Combined streamline upwind Petrov Galerkin method and segregated finite element algorithm for conjugated heat transfer problems, J. Mech. Sci. Technol., 20, 1741-1752 (2009)
[14] Peraire, J.; Peiro, J.; Formaggia, L.; Morgan, K.; Zienkiewica, O. C., Finite element Euler computations in three dimensions, Int. J. Num. Meth. Fluids, 26, 2135-2159 (1995) · Zbl 0665.76073
[15] Peraire, J.; Vahjdati, M.; Morgan, K.; Zienkiewicz, O. C., Adaptive remeshing for compressible flow computation, J. Comput. Phys., 72, 449-466 (1987) · Zbl 0631.76085
[16] Phongthanapanich, S.; Dechaumphai, P., A characteristic-based finite volume element method for convection-diffusion-reaction equation, Trans. Can. Soc. Mech. Eng., 32, 549-559 (2008)
[17] Phongthanapanich, S.; Dechaumphai, P., Combined finite volume and finite element method for convection-diffusion-reaction equation, J. Mech. Sci. Technol., 23, 790-801 (2009)
[18] Phongthanapanich, S.; Dechaumphai, P., Combined finite volume element method for singularly perturbed reaction-diffusion problems, Appl. Math. Comput., 209, 177-185 (2009) · Zbl 1161.65357
[19] Principe, J.; Codina, R., On the stabilization parameter in the subgrid scale approximation of scalar convection-diffusion-reaction equations on distorted meshes, Comput. Meth. Appl. Mech. Eng., 199, 1386-1402 (2010) · Zbl 1227.76039
[20] Wang, H.; Liu, J., Development of CFL-free explicit schemes for multidimensional advection-reaction equations, SIAM J. Sci. Comput., 23, 1418-1438 (2001) · Zbl 1029.65105
[21] Zienkiewicz, O. C.; Codina, R., A general algorithm for compressible and incompressible flow – part I. The split, characteristic-based scheme, Int. J. Num. Meth. Fluids, 20, 869-885 (1995) · Zbl 0837.76043
[22] Zienkiewicz, O. C.; Taylor, R. L., The Finite Element Method Volume 3: Fluid Dynamics (2000), Butterworth-Heinemann · Zbl 0991.74004
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