Operator splitting methods for systems of convection-diffusion equations: nonlinear error mechanisms and correction strategies. (English) Zbl 1051.76046

Summary: Many numerical methods for systems of convection-diffusion equations are based on an operator splitting formulation, where convective and diffusive forces are accounted for in separate substeps. We describe the nonlinear mechanism of the splitting error in such numerical methods in the one-dimensional case, a mechanism that is intimately linked to the local linearizations introduced implicitly in the (hyperbolic) convection steps by the use of an entropy condition. For convection-dominated flows, we demonstrate that operator splitting methods typically generate a numerical widening of viscous fronts, unless the splitting step is of the same magnitude as the diffusion scale. To compensate for the potentially damaging splitting error, we propose a corrected operator splitting (COS) method for general systems of convection-diffusion equations with the ability of correctly resolving the nonlinear balance between the convective and diffusive forces. In particular, COS produces viscous shocks with a correct structure also when the splitting step is large. A front tracking method for systems of conservation laws, which in turn relies heavily on a Riemann solver, constitutes an important part of our COS strategy. The proposed COS method is successfully applied to a system modeling two-phase, multicomponent flow in porous media and a triangular system modeling three-phase flow.


76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
Full Text: DOI


[1] Bressan, A.; Liu, T.-P.; Yang, T., L1 stability estimates for n×n conservation laws, Arch. ration. mech. anal., 149, 1, (1999)
[2] Dahle, H.K., Adaptive characteristic operator splitting techniques for convection – dominated diffusion problems in one and two space dimensions, (1988), University of BergenDepartment of Mathematics Norway
[3] Dahle, H.K.; Espedal, M.S.; Ewing, R.E., Characteristic Petrov-Galerkin subdomain methods for convection-diffusion problems, Numerical simulation in oil recovery, (1986) · Zbl 0699.76099
[4] Dahle, H.K.; Espedal, M.S.; Ewing, R.E., Characteristic Petrov-Galerkin subdomain methods for convection-diffusion problems, Numerical simulation in oil recovery, 77-87, (1988) · Zbl 0699.76099
[5] Dahle, H.K.; Espedal, M.S.; Ewing, R.E.; Sævareid, O., Characteristic adaptive subdomain methods for reservoir flow problems, Numer. meth. partial diff. eqs., 6, 279, (1990) · Zbl 0707.76093
[6] Dahle, H.K.; Espedal, M.S.; Sævareid, O., Characteristic, local grid refinement techniques for reservoir flow problems, Int. J. numer. meth. eng., 34, 1051, (1992) · Zbl 0757.76046
[7] Dahle, H.K.; Ewing, R.E.; Russell, T.F., Eulerian-Lagrangian localized adjoint methods for a nonlinear advection-diffusion equation, Comput. meth. appl. mech. eng., 122, 223, (1995) · Zbl 0851.76058
[8] Dawson, C., Godunov-mixed methods for advection-diffusion equations in multidimensions, SIAM J. numer. anal., 30, 1315, (1993) · Zbl 0791.65062
[9] Dawson, C., High resolution upwind-mixed finite element methods for advection-diffusion equations with variable time-stepping, Numer. meth. partial diff. eqs., 11, 525, (1995) · Zbl 0837.65107
[10] Dawson, C.N., Godunov-mixed methods for advective flow problems in one space dimension, SIAM J. numer. anal., 28, 1282, (1991) · Zbl 0741.65068
[11] Dawson, C.N.; Wheeler, M.F., Time-splitting methods for advection-diffusion-reaction equations arising in contaminant transport, ICIAM 91, Washington, DC, 1991, (1992)
[12] Espedal, M.S.; Ewing, R.E., Characteristic petrov – galerkin subdomain methods for two-phase immiscible flow, Comput. meth. appl. mech. eng., 64, 113, (1987) · Zbl 0607.76103
[13] Espedal, M.S.; Karlsen, K.H., Numerical solution of reservoir flow models based on large time step operator splitting algorithms, Filtration in porous media and industrial applications (cetraro, Italy 1998), 1734, (2000) · Zbl 1077.76546
[14] S. Evje, K. H. Karlsen, K.-A. Lie, and N. H. Risebro, Front tracking and operator splitting for nonlinear degenerate convection – diffusion equations, in Parallel Solution of Partial Differential Equations, edited by P. Bjørstad and M. Luskin, IMA Vol. Math. Appl. Springer-Verlag, New York, 2000, Vol. 120, pp. 209-228. · Zbl 0961.65073
[15] Gimse, T., A numerical method for a system of equations modelling one-dimensional three-phase flow in a porous medium, Nonlinear hyperbolic equations—theory, computation methods, and applications (Aachen, 1988), (1989) · Zbl 0661.76109
[16] Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations, Comm. pure appl. math., 18, 697, (1965) · Zbl 0141.28902
[17] Haugse, V.; Karlsen, K.H.; Lie, K.-A.; Natvig, J.R., Numerical solution of the polymer system by front tracking, Transp. porous media, 44, 63, (2001)
[18] Hoff, D.; Smoller, J., Error bounds for finite-difference approximations for a class of nonlinear parabolic systems, Math. comput., 45, 35, (1985) · Zbl 0613.65096
[19] H. Holden, K. H. Karlsen, K.-A. Lie, and N. H. Risebro. Numerical Solution of Nonlinear Partial Differential Equations using Operator Splitting Methods: An L^{1} Convergence Theory. [Preprint (in preparation)].
[20] Johansen, T.; Winther, R., The solution of the Riemann problem for a hyperbolic system of conservation laws modelling polymer flooding, SIAM J. math. anal., 19, 541, (1988) · Zbl 0658.35061
[21] Karlsen, K.H.; Brusdal, K.; Dahle, H.K.; Evje, S.; Lie, K.-A., The corrected operator splitting approach applied to a nonlinear advection – diffusion problem, Comput. meth. appl. mech. eng., 167, 239, (1998) · Zbl 0942.76047
[22] Karlsen, K.H.; Lie, K.-A., An unconditionally stable splitting scheme for a class of nonlinear parabolic equations, IMA J. numer. anal., 19, 609, (1999) · Zbl 0949.65089
[23] Karlsen, K.H.; Lie, K.-A.; Risebro, N.H.; Frøyen, J., A front-tracking approach to a two-phase fluid-flow model with capillary forces, (special issue on reservoir simulation), 22, 59, (1998)
[24] Karlsen, K.H.; Risebro, N.H., Corrected operator splitting for nonlinear parabolic equations, SIAM J. numer. anal., 37, 1838, (2000)
[25] Karlsen, K.H.; Risebro, N.H., An operator splitting method for convection-diffusion equations, Numer. math., 77, 365, (1997) · Zbl 0882.35074
[26] Kok, J.A., front tracking for three phase flow in porous media, (1994)
[27] K.-A. Lie, A dimensional splitting method for quasilinear hyperbolic equations with variable coefficients, BIT, 39, 683, 1999. · Zbl 0940.65109
[28] Risebro, N.H., A front-tracking alternative to the random choice method, Proc. amer. math. soc., 117, 1125, (1993) · Zbl 0799.35153
[29] Risebro, N.H.; Tveito, A., Front tracking applied to a nonstrictly hyperbolic system of conservation laws, SIAM J. sci. stat. comput., 12, 1401, (1991) · Zbl 0736.65075
[30] Risebro, N.H.; Tveito, A., A front tracking method for conservation laws in one dimension, J. comput. phys., 101, 130, (1992) · Zbl 0756.65120
[31] Tveito, A., Convergence and stability of the Lax-Friedrichs scheme for a nonlinear parabolic polymer flooding problem, Adv. appl. math., 11, 220, (1990) · Zbl 0708.65085
[32] Tveito, A.; Winther, R., The solution of the nonstrictly hyperbolic conservation law may be hard to compute, SIAM J. sci. comp., 16, 320, (1995) · Zbl 0824.65092
[33] Wheeler, M.F.; Kinton, W.A.; Dawson, C.N., Time-splitting for advection-dominated parabolic problems in one space variable, Comm. appl. numer. meth., 4, 413, (1988) · Zbl 0643.65074
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.