Skip to main content Accessibility help

New Splitting Methods for Convection-Dominated Diffusion Problems and Navier-Stokes Equations

  • Feng Shi (a1), Guoping Liang (a2), Yubo Zhao (a3) and Jun Zou (a4)


We present a new splitting method for time-dependent convention-dominated diffusion problems. The original convention diffusion system is split into two sub-systems: a pure convection system and a diffusion system. At each time step, a convection problem and a diffusion problem are solved successively. A few important features of the scheme lie in the facts that the convection subproblem is solved explicitly and multistep techniques can be used to essentially enlarge the stability region so that the resulting scheme behaves like an unconditionally stable scheme; while the diffusion subproblem is always self-adjoint and coercive so that they can be solved efficiently using many existing optimal preconditioned iterative solvers. The scheme can be extended for solving the Navier-Stokes equations, where the nonlinearity is resolved by a linear explicit multistep scheme at the convection step, while only a generalized Stokes problem is needed to solve at the diffusion step and the major stiffness matrix stays invariant in the time marching process. Numerical simulations are presented to demonstrate the stability, convergence and performance of the single-step and multistep variants of the new scheme.


Corresponding author



Hide All
[1]Quarteroni, A. and Valli, A., Numerical Approximation of Partial Differential Equations, Springer-Verlag, Berlin, 1994.
[2]Donea, J. and Huerta, A., Finite Element Methods for Flow Problems, Wiley, New York, 2003.
[3]Glowinski, R. and Tallec, P. Le, Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics, SIAM, Philadelphia, 1989.
[4] T.Hughes, J.R. and Brooks, A.N., A multidimensional upwind scheme with no crosswind diffusion, In T.J.R. Hughes (ed.) Finite Element Methods for Convection Dominated Flows (ASME, New York, 1979) 1935.
[5]Brooks, A.N. and T.Hughes, J.R., Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 32 (1982) 199259.
[6] T.Hughes, J.R., Franca, L. P. and Hulbert, G.M., A new finite element formulation for com-putational fluid dynamics: VIII. The Galerkin/least-squares method for advective-diffusive equations, Comput. Methods Appl. Mech. Engrg. 73 (1989) 173189.
[7]Stynes, M., Steady-state convection-diffusion problems, Acta Numer. 14 (2005) 445508.
[8]John, V. and Novo, J., Error analysis of the SUPG finite element discretization of evolutionary convection-diffusion-reaction equations, SIAM J. Numer. Anal. 49 (2011) 11491176.
[9]Franca, L.P., Frey, S.L. and T.Hughes, J.R., Stabilized finite element methods: I. Application to the advective-diffusive model, Comput. Methods Appl. Mech. Engrg. 96 (1992) 253276.
[10]Franca, L.P. and Frey, S.L., Stabilized finite element methods: II. The incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 99 (1992) 209233.
[11]Franca, L.P. and Farhatb, C., Bubble functions prompt unusual stabilized finite element methods, Comput. Methods Appl. Mech. Engrg. 123 (1995) 299308.
[12]Franca, L.P. and Valentin, F., On an improved unusual stabilized finite element method for the advective-reactive-diffusive equation, Comput. Methods Appl. Mech. Engrg. 190 (2000) 17851800.
[13]Hughes, T.J.R., Multiscale phenomena: Greens functions, the Dirichlet-to-Neumann formulation, subgrid-scale models, bubbles and the origin of stabilized methods, Comput. Methods Appl. Mech. Engrg. 127 (1992) 387401.
[14]Hughes, T.J.R., Feijdo, G.R., Mazzei, L. and Quincy, J.-B., The variational multiscale method - a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg. 166 (1998) 324.
[15]Hughes, T.J.R., Mazzei, L. and Jensen, K.E., The large eddy simulation and the variational multiscale method, Comput. Vis. Sci. 3 (2000) 4759.
[16]John, V., Kaya, S. and Layton, W., A two-level variational multiscale method for convection-dominated convection-diffusion equations, Comput. Methods Appl. Mech. Engrg. 195 (2006) 45944603.
[17]Chen, C.M. and Thomée, V., The lumped mass finite element method for a parabolic problem, J. Austral. Math. Soc. Ser. B 26 (1985) 329354.
[18]Zienkiewicz, O.C. and Codina, R., Search for a general fluid mechanics algorithm, In: Caughey, D.A., Hafez, M.M. (eds.) Frontiers of Computational Fluid Dynamics (Wiley, New York, 1995) 101113.
[19]Zienkiewicz, O.C. and Codina, R., A general algorithm for compressible and incompressible flowlpart I: The split, characteristic-based scheme, Int. J. Numer. Meth. Fluids 20 (1995) 869885.
[20]Zienkiewicz, O.C., Nithiarasu, P., Codina, R., Vázquez, M. and Ortiz, P., The characteristic-based-split procedure: an efficient and accurate algorithm for fluid problems, Int. J. Numer. Meth. Fluids 31 (1999) 359392.
[21]Nithiarasu, P., Zienkiewicz, O.C. and Codina, R., The Characteristic-Based Split (CBS) scheme-a unified approach to fluid dynamics, Int. J. Numer. Meth. Engrg. 66 (2006) 15141546.
[22]Zienkiewicz, O.C., Taylor, R.L. and Nithiarasu, P., The Finite Element Method for Fluid Dynamics (6th Edition), Elsevier, Amsterdam, 2005.
[23]Chorin, A.J., Numerical solution of the Navier-Stokes equations, Math. Comp. 22 (1968) 745762.
[24]Temam, R., Sur l’approximation de la solution des equations de Navier-Stokes par la méthode des fractionnarires II, Arch. Rational Mech. Anal. 33 (1969) 377385.
[25]He, Y., Two-level method based on finite element and Crank-Nicolson extrapolation for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal. 41 (2003), 12631285.
[26]He, Y. and Sun, W., Stability and convergence of the Crank-Nicolson/Adams-Bashforth scheme for the time-dependent Navier-Stokes equations, SIAM J. Numer. Anal. 45 (2007), 837869.
[27]Taylor, C. and Hood, P.,A numerical solution of the Navier-Stokes equations using the finite element technique, Computers & Fluids 1 (1973) 73100.
[28]John, V., Matthies, G. and Rang, J., A comparison of time-discretization/linearization approaches for the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 195 (2006) 59956010.
[29]Ghia, U., Ghia, K.N. and Shin, C.T., High-resolutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J. Comput. Phys. 48 (1982) 387411.
[30]Erturk, E., Corke, T.C. and Gokcol, C., Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, Int. J. Numer. Meth. Fluids 48 (2005) 747774.
[31]Botella, O. and Peyret, R., Benchmark spectral results on the Lid-driven cavity flow, Computers & Fluids, 27 (1998) 421433.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed