×

Parameter estimation in convection dominated nonlinear convection-diffusion problems by the relaxation method and the adjoint equation. (English) Zbl 1138.65086

Summary: The development of numerical methods for strongly nonlinear convection-diffusion problems with dominant convection is an ongoing topic in numerical analysis. For inverse problems in this setting, there is a need of fast and accurate solvers. Here, we present operator splitting with a Riemann solver for the convective part and a relaxation method for the diffusive part, as a means to achieve this goal. Combined with the adjoint equation method this allows us to solve inverse problems within reasonable time frames and with modest computing power. As an example, the dual-well experiment is considered and the adjoint method is compared with a conjugate gradient algorithm and a Levenberg-Marquardt type of iteration method.

MSC:

65M32 Numerical methods for inverse problems for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35R30 Inverse problems for PDEs
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
76S05 Flows in porous media; filtration; seepage
76M20 Finite difference methods applied to problems in fluid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Barrett, J. W.; Knabner, P., Finite element approximation of transport of reactive solutes in porous media, Part 2: error estimates for equilibrium adsorption processes, SIAM J. Numer. Anal., 34, 49-72 (1997)
[2] Constales, D.; Kačur, J.; Malengier, B., A precise numerical scheme for contaminant transport in dual-well flow, Water Resources Res., 39, 30, 1303 (2003)
[3] Crandall, M. G.; Majda, A., The method of fractional steps for conservation laws, Numer. Math., 34, 285-314 (1980) · Zbl 0438.65076
[4] Evje, S., Front tracking and operator splitting for nonlinear degenerate convection-diffusion equations, (Bjørstad, P.; Luskin, M., Parallel Solution of Partial Differential Equations (2000), Springer: Springer Berlin), 209-227 · Zbl 0961.65073
[5] Kačur, J., Solution to strongly nonlinear parabolic problems by a linear approximation scheme, IMA J. Numer. Anal., 19, 119-154 (1999) · Zbl 0946.65145
[7] Kačur, J.; Malengier, B.; Remešíková, M., Solution of contaminant transport with equilibrium and non-equilibrium adsorption, Comput. Methods Appl. Mech. Eng., 194, 2-5, 479-489 (2005) · Zbl 1143.76578
[8] Kačur, J.; Malengier, B.; Van Keer, R., Determination of the diffusion annealing process of Si into Fe, (Neittaanmäki, P.; Rossi, T.; Majava, K.; Pironneau, O., Proceedings of ECCOMAS 2004, vol. 1 (2004), Jyväskylä: Jyväskylä Finland) · Zbl 1391.65165
[9] Karlsen, K. H.; Lie, K.-A., An unconditionally stable splitting for a class of nonlinear parabolic equations, IMA J. Numer. Anal., 19, 4, 609-635 (1999) · Zbl 0949.65089
[10] Remešíková, M., Solution of convection-diffusion problems with non-equilibrium adsorption, J. Comput. Appl. Math., 169, 1, 101-116 (2004) · Zbl 1052.65077
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.