×

zbMATH — the first resource for mathematics

A positivity preserving variational method for multi-dimensional convection-diffusion-reaction equation. (English) Zbl 1380.65272
Summary: A new positivity preserving variational (PPV) procedure is proposed to solve the convection-diffusion-reaction (CDR) equation. Through the generalization of stabilized finite element methods, the present variational procedure offers minimal phase and amplitude errors for different regimes associated with convection, diffusion and reaction effects. By means of Fourier analysis, we first review the shortcomings of the Galerkin/least-squares (GLS) and the subgrid scale (SGS) methods during the change in sign of the reaction coefficient that motivates us for the present linear stabilization as a combined GLS-SGS methodology. Discrete upwind operator with a solution-dependent nonlinear term is then introduced in high gradient regions, which enables the positivity preserving property in the variational formulation. Direct extension to multi-dimensions is carried out by considering the principle streamline and crosswind directions. The efficacy of the method is demonstrated by systematic accuracy and stability analyses in one- and two-dimensions. Results show the reduction of oscillations in the solution in one- and two-dimensional cases and a remarkable reduction in the phase error is observed for the cases with negative reaction coefficient. The proposed formulation provides a superior solution in the reaction-dominated as well as the convection-dominated regimes due to the minimization of spurious oscillations and accurate capturing of the high gradient regions.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35A15 Variational methods applied to PDEs
76R50 Diffusion
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
Software:
SHASTA
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Stynes, M., Steady-state convection-diffusion problems, Acta Numer., 14, 445-508, (2005) · Zbl 1115.65108
[2] Hughes, T. J.R., Finite element methods for convection dominated flows, vol. 34, (1979), ASME New York · Zbl 0418.00017
[3] Brooks, A. N.; Hughes, T. J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 32, 1, 199-259, (1982) · Zbl 0497.76041
[4] Mizukami, A.; Hughes, T. J.R., A Petrov-Galerkin finite element method for convection-dominated flows: an accurate upwinding technique for satisfying the maximum principle, Comput. Methods Appl. Mech. Eng., 50, 2, 181-193, (1985) · Zbl 0553.76075
[5] Kuzmin, D.; Löhner, R.; Turek, S., Flux-corrected transport: principles, algorithms and applications, (2005), Springer
[6] Rice, J. G.; Schnipke, R. J., A monotone streamline upwind finite element method for convection-dominated flows, Comput. Methods Appl. Mech. Eng., 48, 3, 313-327, (1985) · Zbl 0553.76073
[7] LeVeque, R. J., Numerical methods for conservation laws, (1992), Birkhäuser Verlag Basel, Boston · Zbl 0847.65053
[8] Hughes, T. J.R.; Mallet, M.; Mizukami, A., A new finite element formulation for computational fluid dynamics: II. beyond SUPG, Comput. Methods Appl. Mech. Eng., 54, 3, 341-355, (1986) · Zbl 0622.76074
[9] Hughes, T. J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. the generalized streamline operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Eng., 58, 3, 305-328, (1986) · Zbl 0622.76075
[10] Hughes, T. J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Eng., 58, 3, 329-336, (1986) · Zbl 0587.76120
[11] Hughes, T. J.R.; Franca, L. P.; Mallet, M., A new finite element formulation for computational fluid dynamics: VI. convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective-diffusive systems, Comput. Methods Appl. Mech. Eng., 63, 1, 97-112, (1987) · Zbl 0635.76066
[12] Galeão, A. C.; Dutra do Carmo, E. G., A consistent approximate upwind Petrov-Galerkin method for convection-dominated problems, Comput. Methods Appl. Mech. Eng., 68, 1, 83-95, (1988) · Zbl 0626.76091
[13] Dutra do Carmo, E. G.; Alvarez, G. B., A new stabilized finite element formulation for scalar convection-diffusion problems: the streamline and approximate upwind/Petrov-Galerkin method, Comput. Methods Appl. Mech. Eng., 192, 3379-3396, (2003) · Zbl 1054.76055
[14] Hughes, T. J.R.; Franca, L. P.; Hulbert, G. M., A new finite element formulation for computational fluid dynamics: VIII. the Galerkin/least-squares method for advective-diffusive equations, Comput. Methods Appl. Mech. Eng., 73, 2, 173-189, (1989) · Zbl 0697.76100
[15] Franca, L. P.; Dutra do Carmo, E. G., The Galerkin gradient least-squares method, Comput. Methods Appl. Mech. Eng., 74, 41-54, (1989) · Zbl 0699.65077
[16] Shakib, F.; Hughes, T. J.R.; Johan, Z., A new finite element formulation for computational fluid dynamics: X. the compressible Euler and Navier-Stokes equations, Comput. Methods Appl. Mech. Eng., 89, 141-219, (1991) · Zbl 0838.76040
[17] Codina, R., A discontinuity-capturing crosswind-dissipation for the finite element solution of the convection-diffusion equation, Comput. Methods Appl. Mech. Eng., 110, 325-342, (1993) · Zbl 0844.76048
[18] Codina, R., Comparison of some finite element methods for solving the diffusion-convection-reaction equation, Comput. Methods Appl. Mech. Eng., 156, 1, 185-210, (1998) · Zbl 0959.76040
[19] Hughes, T. J.R., Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Eng., 127, 1, 387-401, (1995) · Zbl 0866.76044
[20] Hughes, T. J.R.; Feijóo, G. R.; Mazzei, L.; Quincy, J. B., The variational multiscale method—a paradigm for computational mechanics, Comput. Methods Appl. Mech. Eng., 166, 3-24, (1998) · Zbl 1017.65525
[21] Bazilevs, Y.; Calo, V. M.; Cottrell, J. A.; Hughes, T. J.R.; Reali, A.; Scovazzi, G., Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows, Comput. Methods Appl. Mech. Eng., 197, 1, 173-201, (2007) · Zbl 1169.76352
[22] Akkerman, I.; Bazilevs, Y.; Calo, V. M.; Hughes, T. J.R.; Hulshoff, S., The role of continuity in residual-based variational multiscale modeling of turbulence, Comput. Mech., 41, 3, 371-378, (2008), 2007 · Zbl 1162.76355
[23] Oñate, E., Derivation of stabilized equations for numerical solution of advective-diffusive transport and fluid flow problems, Comput. Methods Appl. Mech. Eng., 151, 233-265, (1998) · Zbl 0916.76060
[24] Oñate, E.; Miquel, J.; Hauke, G., Stabilized formulation for the advection-diffusion-absorption equation using finite calculus and linear finite elements, Comput. Methods Appl. Mech. Eng., 195, 3926-3946, (2006) · Zbl 1178.76234
[25] Oñate, E.; Zarate, F.; Idelsohn, S. R., Finite element formulation for the convective-diffusive problems with sharp gradients using finite calculus, Comput. Methods Appl. Mech. Eng., 195, 1793-1825, (2006) · Zbl 1122.76058
[26] Oñate, E.; Miquel, J.; Zarate, F., Stabilized solution of the multidimensional advection-diffusion-absorption equation using linear finite elements, Comput. Fluids, 36, 92-112, (2007) · Zbl 1123.76034
[27] Boris, J. P.; Book, D. L., Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works, J. Comput. Phys., 11, 38-69, (1973) · Zbl 0251.76004
[28] Zalesak, S. T., Fully multidimensional flux-corrected transport algorithms for fluids, J. Comput. Phys., 31, 335-362, (1979) · Zbl 0416.76002
[29] Harten, A., High resolution schemes for hyperbolic conservation laws, J. Comput. Phys., 49, 357-393, (1983) · Zbl 0565.65050
[30] Tezduyar, T. E.; Park, Y. J., Discontinuity-capturing finite element formulations for nonlinear convection-diffusion-reaction equations, Comput. Methods Appl. Mech. Eng., 59, 307-325, (1986) · Zbl 0593.76096
[31] Idelsohn, S. R.; Nigro, N.; Storti, M.; Buscaglia, G., A Petrov-Galerkin formulation for advective-reaction-diffusion problems, Comput. Methods Appl. Mech. Eng., 136, 27-46, (1996) · Zbl 0896.76042
[32] Harari, I.; Hughes, T. J.R., Stabilized finite element methods for steady advection-diffusion with production, Comput. Methods Appl. Mech. Eng., 115, 165-191, (1994)
[33] Idelsohn, S. R.; Heinrich, J. C.; Oñate, E., Petrov-Galerkin methods for the transient advective-diffusive equation with sharp gradients, Int. J. Numer. Methods Eng., 39, 9, 1455-1473, (1996) · Zbl 0869.76041
[34] Codina, R., On stabilized finite element methods for linear systems of convection-diffusion-reaction equations, Comput. Methods Appl. Mech. Eng., 188, 1, 61-82, (2000) · Zbl 0973.76041
[35] Franca, L. P.; Valentin, F., On an improved unusual stabilized finite element method for the advective-reactive-diffusive equation, Comput. Methods Appl. Mech. Eng., 190, 1785-1800, (2001) · Zbl 0976.76038
[36] Hauke, G.; Garcia-Olivares, A., Variational subgrid scale formulations for the advection-diffusion-reaction equation, Comput. Methods Appl. Mech. Eng., 190, 6847-6865, (2001) · Zbl 0996.76074
[37] Hauke, G., A simple subgrid scale stabilized method for the advection-diffusion-reaction equation, Comput. Methods Appl. Mech. Eng., 191, 2925-2947, (2002) · Zbl 1005.76057
[38] Houzeaux, G.; Eguzkitza, B.; Vázquez, M., A variational multiscale model for the advection-diffusion-reaction equation, Commun. Numer. Methods Eng., 25, 787-809, (2009) · Zbl 1168.65413
[39] Principe, J.; Codina, R., On the stabilization parameter in the subgrid scale approximation of scalar convection-diffusion-reaction equations on distorted meshes, Comput. Methods Appl. Mech. Eng., 199, 21, 1386-1402, (2010) · Zbl 1227.76039
[40] Nadukandi, P.; Oñate, E.; Garcia, J., A high-resolution Petrov-Galerkin method for the 1D convection-diffusion-reaction problem, Comput. Methods Appl. Mech. Eng., 199, 525-546, (2010) · Zbl 1227.76035
[41] Nadukandi, P.; Oñate, E.; Garcia, J., A high-resolution Petrov-Galerkin method for the convection-diffusion-reaction problem. part II—A multidimensional extension, Comput. Methods Appl. Mech. Eng., 213-216, 327-352, (2012) · Zbl 1243.76063
[42] Harari, I.; Hughes, T. J.R., What are C and h?: inequalities for the analysis and design of finite element methods, Comput. Methods Appl. Mech. Eng., 97, 2, 157-192, (1992) · Zbl 0764.73083
[43] Codina, R.; Oñate, E.; Cervera, M., The intrinsic time for the streamline upwind/Petrov-Galerkin formulation using quadratic elements, Comput. Methods Appl. Mech. Eng., 94, 239-262, (1992) · Zbl 0748.76082
[44] Codina, R.; Soto, O., Approximation of the incompressible Navier-Stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes, Comput. Methods Appl. Mech. Eng., 193, 15-16, 1403-1419, (2004) · Zbl 1079.76579
[45] Mittal, S., On the performance of high aspect ratio elements for incompressible flows, Comput. Methods Appl. Mech. Eng., 188, 269-287, (2000) · Zbl 0981.76056
[46] Sengupta, T. K.; Dipankar, A.; Sagaut, P., Error dynamics: beyond von Neumann analysis, J. Comput. Phys., 226, 2, 1211-1218, (2007) · Zbl 1125.65337
[47] Sengupta, T. K., High accuracy computing methods: fluid flows and wave phenomena, (2013), Cambridge University Press · Zbl 06218355
[48] Warming, R. F.; Hyett, B. J., The modified equation approach to the stability and accuracy analysis of finite-difference methods, J. Comput. Phys., 14, 159-179, (1974) · Zbl 0291.65023
[49] Codina, R., A finite element formulation for the numerical solution of the convection-diffusion equation, (1993), CIMNE Barcelona · Zbl 1003.65504
[50] M. Casey, T. Wintergerste, Best practice guidelines for industrial computational fluid dynamics, ERCOFTAC, 2000.
[51] Christon, M. A.; Martinez, M. J.; Voth, T. E., Generalized Fourier analyses of the advection-diffusion equation—part I: one-dimensional domains, Int. J. Numer. Methods Fluids, 45, 8, 839-887, (2004) · Zbl 1085.76054
[52] Hauke, G.; Doweidar, M. H., Fourier analysis of semi-discrete and space-time stabilized methods for the advective-diffusive-reactive equation: II. SGS, Comput. Methods Appl. Mech. Eng., 194, 691-725, (2005) · Zbl 1112.76390
[53] Hauke, G.; Doweidar, M. H., Fourier analysis of semi-discrete and space-time stabilized methods for the advective-diffusive-reactive equation: I. SUPG, Comput. Methods Appl. Mech. Eng., 194, 45-81, (2005) · Zbl 1112.76391
[54] Shakib, F.; Hughes, T. J.R., A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space-time Galerkin/least-squares algorithms, Comput. Methods Appl. Mech. Eng., 87, 35-58, (1991) · Zbl 0760.76051
[55] Sengupta, T. K.; Bhumkar, Y. G.; Rajpoot, M. K.; Suman, V. K.; Saurabh, S., Spurious waves in discrete computation of wave phenomena and flow problems, Appl. Math. Comput., 218, 9035-9065, (2012) · Zbl 1245.65112
[56] Voth, T. E.; Martinez, M. J.; Christon, M. A., Generalized Fourier analyses of the advection-diffusion equation—part II: two-dimensional domains, Int. J. Numer. Methods Fluids, 45, 8, 889-920, (2004) · Zbl 1085.76059
[57] Sengupta, T. K.; Rajpoot, M. K.; Saurabh, S.; Vijay, V. V.S. N., Analysis of anisotropy of numerical wave solutions by high accuracy finite difference methods, J. Comput. Phys., 230, 27-60, (2011) · Zbl 1205.65239
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.