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On the solution of the advection equation and advective dominated reactor models by weighted residual methods. (English) Zbl 1390.76625

Summary: The least-squares, tau and orthogonal collocation methods are employed to give approximate solutions to, e.g., the advective equation and an advective dominated fixed packed bed reactor model. It has been argued in the literature that the least-squares formulation is a unified method for the approximate solution of differential equations governing diverse areas in engineering and science. In particular, it has been argued that the least-squares method is suited for first-order differential operators. To investigate this, five numerical test problems have been constructed. The least-squares method showed rather poor numerical accuracy for the solution of a chemical process operated in a fixed packed bed reactor. The classical orthogonal collocation and tau formulations showed superior numerical accuracy to that of the least-squares method for this problem. For other test cases, i.e., first-order ordinary differential equation and finite-element solution of a one-dimensional advection equation with a sharp gradient in the solution, the least-squares method obtained better numerical results than the orthogonal collocation and tau methods. The least-squares method is not necessarily the better choice for advection dominated problems as a number of different problems favor the orthogonal collocation and/or tau methods. However, for the advective terms of some problems the least-squares method can be used to stabilize the Galerkin discretization. The choice of the better approximation method for an advection dominated problem is case dependent.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
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[1] Tannehill, J. C.; Anderson, D. A.; Pletcher, R. H., Computational fluid mechanics and heat transfer, (1997), Taylor & Francis Philadelphia
[2] Versteeg, H. K.; Malalasekera, W., Computational fluid dynamics. The finite volume method, (2007), Pearson England
[3] Crandall, S. H., Engineering analysis, (1965), McGraw-Hill New York
[4] Villadsen, J. V.; Steward, W. E., Solution of boundary-value problems by orthogonal collocation, Chem Eng Sci, 22, 1482-1501, (1967)
[5] Galerkin, B. G., Series solution of some problems in elastic equilibrium of rods and plates, Vestn Inzh Tech, 19, 897-908, (1915)
[6] Lanczos, C., Trignometric interpolation of empirical and analytical functions, J Math Phys, 17, 123-199, (1938) · JFM 64.0577.02
[7] Jiang, B.-N., The least-squares finite element method: theory and applications in computational fluid dynamics and electromagnetics, (1998), Springer Berlin
[8] Dziskariani, A. V., The least-squares and Bubnov-Galerkin methods, Z Vycisl Mat Mat Fiz, 8, 1110-1160, (1968) · Zbl 0182.21202
[9] Lucka, A., The rate of convergence to zero of the residual and the error in the Bubnov-Galerkin method and the method of least-squares, Proceedings of seminar on differential and integral equations, No 1, pp. 113-122 (Russian), Akad. Nauk Ukrain, SSR Inst. Mat., Kiev, Ukrain, (1969)
[10] Bramble, J. H.; Schatz, A., Least-squares for 2mth-order elliptic boundary-value problems, Math Comput, 25, 1-32, (1971) · Zbl 0216.49202
[11] Lynn, P. P.; Arya, S. K., Finite elements formulation by the weighted discrete least-squares method, Int J Numer Methods Eng, 8, 71-90, (1974) · Zbl 0272.65103
[12] Zienkiewicz, O., Least squares finite element for elasto-static problems - use of reduced integration, Int J Numer Methods Eng, 8, 341-358, (1974) · Zbl 0276.73040
[13] Chang, C. L.; Nelson, J. J., Least-squares finite element method for the Stokes problem with zero residual of mass conservation, Soc Ind Appl Math, 34, 480-489, (1997) · Zbl 0890.76036
[14] Proot, M. M.J.; Gerritsma, M. I., Least-squares spectral element applied to the Stokes problem, J Comput Phys, 181, 454-477, (2002) · Zbl 1178.76270
[15] Proot, M. M.J.; Gerritsma, M. I., A least-squares spectral element formulation for Stokes problem, J Sci Commut, 17, 1-4, 285-296, (2002) · Zbl 1036.76046
[16] Proot, M. M.J.; Gerritsma, M. I., Mass- and momentum conservation of the least-squares spectral element method for the Stokes problem, J Sci Commut, 27, 1-3, 389-401, (2006) · Zbl 1100.76049
[17] Gerritsm, M.; van der Bas, R.; Maerschalck, B. D.; Koren, B.; Deconinck, H., Least-squares spectral element method applied to the Euler equations, Int J Numer Methods Fluids, 57, 1371-1395, (2008) · Zbl 1140.76030
[18] Reddy, J. N.; Pontaza, J. P., Spectral h/p least-squares finite element formulation for the Navier-Stokes equations, J Comput Phys, 190, 523-549, (2003) · Zbl 1077.76054
[19] Reddy, J. N.; Pontaza, J. P., Space-time coupled spectral/hp least squares finite element formulation for the incompressible Navier-Stokes equation, J Comput Phys, 197, 418-459, (2004) · Zbl 1106.76403
[20] Pontaza, J. P., A new consistent splitting scheme for incompressible Navier-Stokes flows: a least-squares spectral element implementation, J Comput Phys, 225, 1590-1602, (2007) · Zbl 1343.76020
[21] Pontaza, J. P., A least-squares finite element formulation for unsteady incompressible flows with improved velocity-pressure coupling, J Comput Phys, 217, 536-588, (2006) · Zbl 1142.76033
[22] Mazzarella, G.; Maggio, F.; Pizianti, C., Least-squares spectral element method for the 2D Maxwell equations in frequency domain, Int J Numer Model, 17, 6, 509-522, (2004) · Zbl 1161.78336
[23] Gerritsma, M., Direct minimization of the discontinuous least-squares spectral element method for viscoelastic fluids, J Sci Comput, 27, 245-256, (2006) · Zbl 1101.76040
[24] De Maerschalck, B.; Gerritsma, M. I., Least-squares spectral element method for non-linear hyperbolic differential equations, J Comput Appl Math, 215, 357-367, (2008) · Zbl 1138.65088
[25] Lin, W. H., A least-squares Legendre spectral element method for sound propagation problems, J Acoust Soc Am, 104, 5, 3111-3114, (1998)
[26] Lin, W. H., A least-squares Legendre spectral element solution to sound propagation problems, J Acoust Soc Am, 109, 2, 465-474, (2001)
[27] Dorao, C. A., High order methods for the solution of the population balance equation with applications to bubbly flows, (2006), Norwegian University of Science and Technology (NTNU), [Ph.D. thesis]
[28] Zhu, Z., The least-squares spectral element method solution of the gas-liquid multi-fluid model coupled with the population balance equation, (2009), Norwegian University of Science and Technology (NTNU), [Ph.D. thesis]
[29] Patruno, L. E., Experimental and numerical investigations of liquid fragmentation and droplet generation for gas processing at high pressures, (2010), Norwegian University of Science and Technology (NTNU), [Ph.D. thesis]
[30] Nayak, A. K.; Borka, Z.; Patruno, L. E.; Sporleder, F.; Dorao, C. A.; Jakobsen, H. A., A combined multifluid-population balance model for vertical gas-liquid bubble-driven flows considering bubble column operating conditions, Ind Eng Chem Res, 50, 1786-1798, (2011)
[31] Sporleder, F., Simulation of chemical reactors using the least-squares spectral element method, (2011), Norwegian University of Science and Technology (NTNU), [Ph.D. thesis]
[32] Rout, K. R., A study of the sorption enhanced steam methane reforming process, (2012), Norwegian University of Science and Technology (NTNU), [Ph.D. thesis]
[33] Carey, G. F.; Jiang, B. N., Least-squares finite elements for first-order hyperbolic systems, Int J Numer Methods Eng, 26, 81-93, (1988) · Zbl 0641.65080
[34] 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, 199-259, (1982) · Zbl 0497.76041
[35] Kumar, R.; Dennis, B. H., Bubble-enriched least-squares finite element method for transient advective transport, Differ Equ Nonlinear Mech, 2008, (2008), ID 267454, 21 pages · Zbl 1163.65068
[36] Jiang, B. N.; Povinelli, L. A., Least-squares finite element method for fluid dynamics, Comput Methods Appl Mech Eng, 81, 13-37, (1990) · Zbl 0714.76058
[37] Dorao, C. A.; Jakobsen, H. A., A parallel time-space least squares spectral element solver for incompressible flow problems, Appl Math Comput, 185, 45-58, (2007) · Zbl 1108.76052
[38] Solsvik, J.; Becker, P. J.; Sheibat-Othman, N.; Jakobsen, H. A., On the solution of the dynamic population balance model describing emulsification: evaluation of weighted residual methods, Can J Chem Eng, 92, 250-265, (2014)
[39] Solsvik, J.; Jakobsen, H. A., Evaluation of the weighted residual methods for the solution of a population balance model describing bubbly flows: the least-squares, Galerkin, tau and orthogonal collocation methods, Ind Eng Chem Res, 52, 15988-16013, (2013)
[40] Solsvik, J.; Jakobsen, H. A., Effect of Jacobi polynomials on the numerical solution of the Pellet equation using the orthogonal collocation, Galerkin, tau and least squares methods, Comput Chem Eng, 39, 1-21, (2012)
[41] Solsvik, J.; Jakobsen, H. A., Evaluation of spectral, spectral-element and finite-element methods for the solution of the Pellet equation, Can J Chem Eng, 92, 1396-1413, (2014)
[42] Solsvik, J.; Tangen, S.; Jakobsen, H. A., Evaluation of weighted residual methods for the solution of the Pellet equations: the orthogonal collocation, Galerkin, tau and least-squares methods, Comput Chem Eng, 58, 223-259, (2013)
[43] Hughes, T. J.R.; Franca, L. P.; Hulbert, H. 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, 173-189, (1989) · Zbl 0697.76100
[44] Jiang, B. N.; Lin, T. L.; Povinelli, L. A., Large-scale computation of incompressible viscous flow by least-squares finite element method, Comput Methods Appl Mech Eng, 114, 213-231, (1994)
[45] Hughes, T. J.R.; Brooks, A. N., A theoretical framework for Petrov-Galerkin methods with discontinuous weighting functions: application to the streamline upwind procedure, (Gallagher, R. H.; Norrie, D. H.; Oden, J. T.; Zienkiewicz, O. C., Finite elements in fluids, vol. IV, (1982), Wiley London), 46-65
[46] Hughes, T. J.R.; Brooks, A. N., A multidimensional upwind scheme with no crosswind diffusion, (Hughes, T. J.R., (1979), ASME New York), 19-35 · Zbl 0423.76067
[47] Bochev, P. B.; Choi, J., A comparative study of least-squares, SUPG and Galerkin methods for convection problems, Int J Comput Fluid Dyn, 15, 127-146, (2001) · Zbl 0982.76056
[48] John, V.; Schmeyer, E., Finite element methods for time-dependent convection-diffusion-reaction equations with small diffusion, Comput Methods Appl Mech Eng, 198, 475-494, (2008) · Zbl 1228.76088
[49] Proot, M. M.J., The least-squares spectral element method, (2003), Delt University of Technology, The Netherlands, [Ph.D. thesis] · Zbl 1153.76399
[50] Gerritsma, M. I.; De Maerschalck, B., Advanced computational methods in science and engineering, Lecture notes in computational science and engineering, vol. 71, 179-227, (2010), Springer-Verlag Berlin · Zbl 1423.76322
[51] Szegö, G., Orthogonal polynomials, (1939), American Mathematical Society New York · JFM 65.0286.02
[52] Golub, G. H.; Welsch, J. H., Calculation of Gauss quadrature rules, Math Comput, 23, 221-230, (1969) · Zbl 0179.21901
[53] Golub, G. H., Some modified matrix eigenvalue problems, Siam Rev, 15, 318-334, (1973) · Zbl 0254.65027
[54] Solsvik, J.; Jakobsen, H. A., On the solution of the population balance equation for bubbly flows using the high-order least-squares method: implementation issues, Rev Chem Eng, 29, 63-98, (2013)
[55] Solsvik, J.; Jakobsen, H. A., Solution of the dynamic population balance equation describing breakage-coalescence systems in agitated vessels: the least-squares method, Can J Chem Eng, 92, 268-287, (2014)
[56] Jakobsen, H. A., Numerical convection algorithms and their role in Eulerian CFD reactor simulations, Int J Chem React Eng, 1, 1-17, (2003)
[57] Bartnicki, J.; Olendrzynski, K.; Abert, K.; Seibert, P.; Morariu, B., Numerical approximation of the transport equation: comparison of five positive definite algorithms, (1990), International Institute for Applied System Analysis, IIASA Laxenburg, Austria
[58] Froment, G. F.; Bischoff, K. B., Chemical reactor analysis and design, (1990), John Wiley & Sons Hoboken, NJ
[59] Maxwell, J. C., On the dynamical theory of gases, Philos Trans R Soc, 157, 49-88, (1866)
[60] Stefan, J., Über das gleichgewicht und die bewegung insbesondere die diffusion von gasgemengen, Sitzber Akad Wiss Wien, 63, 63-124, (1871)
[61] Krishna, R.; Wesselingh, J. A., The Maxwell-Stefan approach to mass transfer, Chem Eng Sci, 52, 6, 861-911, (1996)
[62] Solsvik, J.; Jakobsen, H. A., A survey of multicomponent mass diffusion flux closures for porous pellets: mass and molar forms, Transp Porous Media, 93, 99-126, (2012)
[63] Jakobsen, H. A., Chemical reactor modeling: multiphase reactive flows, (2008), Springer Berlin
[64] Xu, J.; Froment, G. F., Methane steam reforming. II. diffusional limitations and reactor simulation, AIChE J, 35, 88-96, (1989)
[65] Ferziger, J. H.; Peric, M., Computational methods for fluid dynamics, (2002), Springer Berlin · Zbl 0998.76001
[66] De Maerschalck, B.; Gerritsma, M. I.; Proot, M. M.J., Space-time least-squares spectral elements for convection-dominated unsteady flows, AIAA J, 44, 558-565, (2006)
[67] Sporleder, F.; Dorao, C. A.; Jakobsen, H. A., Simulation of chemical reactors using the least-squares spectral element method, Chem Eng Sci, 65, 5146-5159, (2010)
[68] Rout, K. R.; Jakobsen, H. A., Reactor performance optimization by the use of a novel combined Pellet reflecting both catalyst and adsorbent properties, Fuel Process Technol, 99, 13-34, (2012)
[69] De Maerschalck, B.; Gerritsma, M. I., Higher-order Gauss-lobatto integration for non-linear hyperbolic equations, J Sci Comput, 27, 201-214, (2006) · Zbl 1115.65103
[70] Solsvik, J., Chemical reactor investigations: modeling, implementation and simulation, (2014), Norwegian University of Science and Technology (NTNU), [Ph.D. thesis]
[71] Hughes, T.; Franca, L. P.; Hulbert, G. M., A new finite element formulation for computational fluid dynamics: V. circumventing the babuska-Brezzi condition: a stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations, Comput Methods Appl Mech Eng, 59, 85-99, (1986) · Zbl 0622.76077
[72] Franca, L. P.; Frey, S. L.; Hughes, T., Stabilized finite element methods: I. application to the advection-diffusive model, Comput Methods Appl Mech Eng, 95, 253-276, (1992) · Zbl 0759.76040
[73] Camprub, N.; Colominas, I.; Navarrina, F.; Casteleiro, M., Galerkin, least-squares and G.L.S numerical approaches for convective-diffusive transport problems in engineering, Proceedings of European congress on computational methods in applied science and engineering, Barcelona, (2000)
[74] Donea, J.; Huerta, A., Finite element methods for flow problems, (2003), Wiley NJ, USA
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