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Numerical artifacts in the generalized porous medium equation: why harmonic averaging itself is not to blame. (English) Zbl 1391.76734
Summary: The degenerate parabolic generalized porous medium equation (GPME) poses numerical challenges due to self-sharpening and its sharp corner solutions. For these problems, we show results for two subclasses of the GPME with differentiable \(k(p)\) with respect to \(p\), namely the porous medium equation (PME) and the superslow diffusion equation. Spurious temporal oscillations, and nonphysical locking and lagging have been reported in the literature. These issues have been attributed to harmonic averaging of the coefficient \(k(p)\) for small \(p\), and arithmetic averaging has been suggested as an alternative. We show that harmonic averaging is not solely responsible and that an improved discretization can mitigate these issues. Here, we investigate the causes of these numerical artifacts using modified equation analysis. The modified equation framework can be used for any type of discretization. We show results for the second order finite volume method. The observed problems with harmonic averaging can be traced to two leading error terms in its modified equation. This is also illustrated numerically through a modified harmonic method (MHM) that can locally modify the critical terms to remove the aforementioned numerical artifacts.

MSC:
76S05 Flows in porous media; filtration; seepage
76M25 Other numerical methods (fluid mechanics) (MSC2010)
35K65 Degenerate parabolic equations
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[1] Lipnikov, K.; Manzini, G.; Moulton, D.; Shashkov, M., The mimetic finite difference method for elliptic and parabolic problems with a staggered discretization of diffusion coefficient, J. Comput. Phys., 305, 111-126, (2016) · Zbl 1349.65315
[2] Kadioglu, S.; Nourgaliev, R.; Mousseau, V., A comparative study of the harmonic and arithmetic averaging of diffusion coefficients for non-linear heat conduction problems, 1-25, (2008), Idaho National Laboratory
[3] Nie, C.; Yu, H., A novel finite volume scheme with geometric average method for radiative heat transfer problems, Appl. Phys. Front., 1, 4, 32-44, (2013)
[4] van der Meer, J.; Kraaijevanger Möller, J.; Jansen, J., Temporal oscillations in the simulation of foam enhanced oil recovery, (ECMOR XV - 15th European Conference on the Mathematics of Oil Recovery, (2016)), 1-20
[5] D. Maddix, L. Sampaio, M. Gerritsen, Harmonic versus arithmetic averaging in reservoir modeling, in preparation. · Zbl 1391.76734
[6] Vázquez, J., The porous medium equation: mathematical theory, (2007), The Clarendon Press, Oxford University Press Oxford · Zbl 1107.35003
[7] Ngo, C.; Huang, W., A study on moving mesh finite element solution of the porous medium equation, J. Comput. Phys., 331, 357-380, (2017) · Zbl 1378.76110
[8] Zel’dovich, Y. B.; Raizer, Y. P., Physics of shock waves and high-temperature hydrodynamic phenomena II, (1966), Academic Press New York
[9] Boussinesq, J., Recherches théoriques sue l’écoulement des nappes d’eau infiltrés dans le sol et sur le débit de sources, Comptes Rendus Acad. Sci./J. Math. Pures Appl., 10, 5-78, (1903/1904) · JFM 35.0761.01
[10] Galaktionov, V.; Vázquez, J., Equation of superslow diffusion, (A Stability Technique for Evolution Partial Differential Equations: A Dynamical Systems Approach, vol. 56, (2004)), 57-79
[11] Galaktionov, V.; Vázquez, J., Asymptotic behavior for an equation of superslow diffusion. the Cauchy problem, (1991)
[12] Galaktionov, V.; Svirshchevskiı̌, S., Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics, Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series, (2007), Chapman and Hall/CRC, Taylor and Francis Group Boca Raton · Zbl 1153.35001
[13] Maddix, D.; Gerritsen, M.; Sampaio, L.; Nissen, A., Numerical artifacts in the discontinuous generalized porous medium equation: how to avoid spurious temporal oscillations, J. Comput. Phys., 361, 280-298, (2018) · Zbl 1392.76038
[14] Barenblatt, G.; Vishik, M., On finite velocity of propagation in problems of non-stationary filtration of a liquid of gas, Prikl. Mat. Mech., 20, 411-417, (1956), (in Russian) · Zbl 0074.43006
[15] Barenblatt, G.; Zel’dovich, Y., The asymptotic properties of self-modeling solutions of the non-stationary gas filtration equations, Sov. Phys. Dokl., 3, 44-47, (1958)
[16] Oleĭnik, O.; Kalašinkov, A.; Čžou, Y., The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR, Ser. Mat., 22, 667-704, (1958) · Zbl 0093.10302
[17] Kalašinkov, A., Formation of singularities in solutions of the equation of non stationary filtration, Ž. Vyčisl. Mat. Mat. Fiz., 7, 440-444, (1967)
[18] Aronson, D., Properties of flows through porous media, SIAM J. Appl. Math., 17, 2, 461-467, (1969) · Zbl 0187.03401
[19] Shmarev, S., Interfaces in multidimensional diffusion equations with absorption terms, Nonlinear Anal., 53, 791-828, (2003) · Zbl 1031.35082
[20] Shmarev, S., Interfaces in solutions of diffusion-absorption equations in arbitrary space dimension, (Progress in Nonlinear Differential Equations and Their Applications, vol. 61, (2005)), 257-273 · Zbl 1083.35028
[21] Galaktionov, V.; Samarskiı̌, A., Methods of constructing approximate self-similar solutions of nonlinear heat equations. I, Math. USSR Sb., 46, 291-321, (1983) · Zbl 0529.35043
[22] Zhang, Q.; Wu, Z., Numerical simulation for porous medium equation by local discontinuous Galerkin finite element method, J. Sci. Comput., 38, 127-148, (2009) · Zbl 1203.65193
[23] Lax, P.; Wendroff, B., Systems of conservation laws, Commun. Pure Appl. Math., XIII, 217-237, (1960) · Zbl 0152.44802
[24] Gottlieb, S.; Shu, C., Total variation diminishing Runge-Kutta schemes, Math. Comput., 67, 221, 73-85, (1998) · Zbl 0897.65058
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