Un nuevo esquema iterativo robusto y eficiente para un problema parabólico no lineal degenerado.
Resumen
El flujo de agua a través del suelo esta modelado matemáticamente por la ecuación de Richards. Esta ecuación doblemente no lineal degenerada, es difícil de resolver, más aun la no linealidad y la degeneración hacen que el diseño de esquemas numéricos para este problema sea una tarea desafiante. Según la literatura los métodos implícitos son los que dan mejores resultados ya que los esquemas obtenidos permiten simular el problema degenerado sin embargo estos producen problemas no lineales que deben ser resueltos mediante métodos de linealización.
En esta investigación se desarrolla un nuevo esquema numérico de linealización de la ecuación de Richards. Se usa un esquema de Euler totalmente implícito para discretizar el tiempo y elementos finitos para discretizar el espacio, sin embargo, estos esquemas deberían funcionar para cualesquier discretización espacial escogida. Se consiguió un esquema mas robusto y más rápido (en lo que a número de iteraciones y tiempo total de máquina empleado se refiere) que los ya existentes.
Citas
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Bergamaschi, L., & Putti, M. (1999). Mixed finite elements and newton type linearizations for the solution of richards equation. International Journal for Numerical Methods in Engineering, 45, 1025-1046.
Berninger, H., Kornhuber, R., & Sander, O. (2011). Fast and robust numerical solution of the richards equation in homogeneous soil. SIAM Journal on Numerical Analysis, 49, 2576–2597.
Brenner, K., & Cancès, C. (2017). Improving newtons method performance by parametrization: the case of the richards equation. SIAM Journal on NumericalAnalysis, 55, 1760-1785.
Brenner, K., Hilhorst, D., & Vu-Do, H. (2016). The generalized finite volumen sushi scheme for the discretization of richards equation. Vietnam Journal of Mathematics, 44, 557-586.
Buckingham, E. (1907). Studies on the movement of soil moisture. U.S. Department of Agriculture Bureau of Soils, 38, 0.
Cai, X. C., & Keyes, D. E. (2002). Nonlinearly preconditioned inexact newton algorithms. Siam Journal on Scientific Computing, 24, 183-200.
Caviedes, D., P., G., & Murillo, J. (2013). conservation, stability and efficiency of a finite volume method for the 1d richards equation. Journal of Hydrology, 480, 69-84.
Celia, M., & Bouloutas, E. (1990). General mass-conservative numerical solutions for the unsaturated flow equation. Water Resources Research, 26, 1483-1496.
Dolean, V., Gander, M. J., Kheriji, W., Kwok, F., & R., M. (2016). Nonlinear preconditioning: How to use a nonlinear schwarz method to precondition newton’s method. SIAM J. Scientific Computing, 6, 3357–3380.
Eymard, R., Gutnic, M., & Hilhorst, D. (1999). The finite volume method for richards equation. Computational Geosciences, 3, 259-294.
Knabner, P., & Angermann, L. (2003). Numerical methods for elliptic and parabolic partial differential equations. Springer.
Kuraz, M., Mayer, P., & Pech, P. (2015). Solving the nonlinear and nonstationary richards equation with two-level adaptive domain decomposition (ddadaptivity). Applied Mathematics and Computation, 267, 207-222.
Lai, W., & Ogden, F. (2015). A mass-conservative finite volume predictor–corrector solution of the 1d richards’ equation. Journal of Hydrology, 523, 119–127.
Lehmann, F., & Ackerer, P. (1998). Comparison of iterative methods for improved solutions of the fluid flow equation in partially saturated porous media. Transport in Porous Media, 31, 275-292.
Lipnikova, K., Moultona, D., & Svyatskiy, D. (en prensa). New preconditioning strategy for jacobian-free solvers for variably saturated flows with. Advances in Water Resources, 0, 0.
List, F., & Radu, F. (2016). study on iterative methods for solving richards equation. Computational Geosciences, 20, 341-35.
Marinoschi, G. (2010). Functional approach to nonlinear models of water flow in soils (1st Edition. ed.). Springer.
Misiats, O., & Lipnikov, K. (2013). Second-order accurate monotone finite volumen scheme for richards equation. Journal of Computational Physics, 239, 123137.
Pop, I., & Radu, F. (2004). Mixed fnite elements for the richards equation: linearization procedure. Journal of Computational and Applied Mathematics, 168,365-373.
Radu, F., Pop, I., & Knabner, P. (2004). Order of convergence estimates for an euler implicit, mixed finite element discretization of richards’ equation. SIAM Journal on Numerical Analysis, 42, 1452-1478.
Radu, F., Pop, I., & Knabner, P. (2006). On the convergence of the newton method for the mixed finite element discretization of a class of degenerate parabolic equation. Numerical Mathematics and Advanced Applications, 42, 1194–1200.
Radu, F., Pop, I., & Knabner, P. (2008). Error estimates for a mixed finite element discretization of some degenerate parabolic equations. Numerische Mathematik, 109, 285-311.
Raviart, P., & Thomas, J. (1977). A mixed finite element method for second order elliptic problems. in mathematical aspects of the finite elements method, galligani i, magenes e (eds) lecture notes
in mathematics. Springer:New York,, 606.
Richards, L. (1931). Capillary conduction of liquids through porous mediums. Journal of Applied Physics, 1, 318-333.
Slodicka, M. (2002). A robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic problems arising in flow in porous media. Siam Journal on Scientific Computing, 23, 1593-1614.
W., J., & Delleur. (2006). The handbook of groundwater engineering, second edition. CRC Press.
Wang, X., & H., T. (2013). Trust-region based solver for nonlinear transport in heterogeneous porous media. J. Comput. Phys., 253, 114–137.
Younis, R., Tchelepi, A., Dagmar, T., & Aziz, K. (2010). Adaptively localized continuation-newton method nonlinear solvers that converge all the time. SPE Journal, 15, 526-544.
Zambra, C., Dumbser, M., Toro, E., & Moraga, N. (2012). A novel numerical method of high-order accuracy for flow in unsaturated porous media. International Journal for Numerical Methods in Engineering, 89, 227-240.
Bergamaschi, L., & Putti, M. (1999). Mixed finite elements and newton type linearizations for the solution of richards equation. International Journal for Numerical Methods in Engineering, 45, 1025-1046.
Berninger, H., Kornhuber, R., & Sander, O. (2011). Fast and robust numerical solution of the richards equation in homogeneous soil. SIAM Journal on Numerical Analysis, 49, 2576–2597.
Brenner, K., & Cancès, C. (2017). Improving newtons method performance by parametrization: the case of the richards equation. SIAM Journal on NumericalAnalysis, 55, 1760-1785.
Brenner, K., Hilhorst, D., & Vu-Do, H. (2016). The generalized finite volumen sushi scheme for the discretization of richards equation. Vietnam Journal of Mathematics, 44, 557-586.
Buckingham, E. (1907). Studies on the movement of soil moisture. U.S. Department of Agriculture Bureau of Soils, 38, 0.
Cai, X. C., & Keyes, D. E. (2002). Nonlinearly preconditioned inexact newton algorithms. Siam Journal on Scientific Computing, 24, 183-200.
Caviedes, D., P., G., & Murillo, J. (2013). conservation, stability and efficiency of a finite volume method for the 1d richards equation. Journal of Hydrology, 480, 69-84.
Celia, M., & Bouloutas, E. (1990). General mass-conservative numerical solutions for the unsaturated flow equation. Water Resources Research, 26, 1483-1496.
Dolean, V., Gander, M. J., Kheriji, W., Kwok, F., & R., M. (2016). Nonlinear preconditioning: How to use a nonlinear schwarz method to precondition newton’s method. SIAM J. Scientific Computing, 6, 3357–3380.
Eymard, R., Gutnic, M., & Hilhorst, D. (1999). The finite volume method for richards equation. Computational Geosciences, 3, 259-294.
Knabner, P., & Angermann, L. (2003). Numerical methods for elliptic and parabolic partial differential equations. Springer.
Kuraz, M., Mayer, P., & Pech, P. (2015). Solving the nonlinear and nonstationary richards equation with two-level adaptive domain decomposition (ddadaptivity). Applied Mathematics and Computation, 267, 207-222.
Lai, W., & Ogden, F. (2015). A mass-conservative finite volume predictor–corrector solution of the 1d richards’ equation. Journal of Hydrology, 523, 119–127.
Lehmann, F., & Ackerer, P. (1998). Comparison of iterative methods for improved solutions of the fluid flow equation in partially saturated porous media. Transport in Porous Media, 31, 275-292.
Lipnikova, K., Moultona, D., & Svyatskiy, D. (en prensa). New preconditioning strategy for jacobian-free solvers for variably saturated flows with. Advances in Water Resources, 0, 0.
List, F., & Radu, F. (2016). study on iterative methods for solving richards equation. Computational Geosciences, 20, 341-35.
Marinoschi, G. (2010). Functional approach to nonlinear models of water flow in soils (1st Edition. ed.). Springer.
Misiats, O., & Lipnikov, K. (2013). Second-order accurate monotone finite volumen scheme for richards equation. Journal of Computational Physics, 239, 123137.
Pop, I., & Radu, F. (2004). Mixed fnite elements for the richards equation: linearization procedure. Journal of Computational and Applied Mathematics, 168,365-373.
Radu, F., Pop, I., & Knabner, P. (2004). Order of convergence estimates for an euler implicit, mixed finite element discretization of richards’ equation. SIAM Journal on Numerical Analysis, 42, 1452-1478.
Radu, F., Pop, I., & Knabner, P. (2006). On the convergence of the newton method for the mixed finite element discretization of a class of degenerate parabolic equation. Numerical Mathematics and Advanced Applications, 42, 1194–1200.
Radu, F., Pop, I., & Knabner, P. (2008). Error estimates for a mixed finite element discretization of some degenerate parabolic equations. Numerische Mathematik, 109, 285-311.
Raviart, P., & Thomas, J. (1977). A mixed finite element method for second order elliptic problems. in mathematical aspects of the finite elements method, galligani i, magenes e (eds) lecture notes
in mathematics. Springer:New York,, 606.
Richards, L. (1931). Capillary conduction of liquids through porous mediums. Journal of Applied Physics, 1, 318-333.
Slodicka, M. (2002). A robust and efficient linearization scheme for doubly nonlinear and degenerate parabolic problems arising in flow in porous media. Siam Journal on Scientific Computing, 23, 1593-1614.
W., J., & Delleur. (2006). The handbook of groundwater engineering, second edition. CRC Press.
Wang, X., & H., T. (2013). Trust-region based solver for nonlinear transport in heterogeneous porous media. J. Comput. Phys., 253, 114–137.
Younis, R., Tchelepi, A., Dagmar, T., & Aziz, K. (2010). Adaptively localized continuation-newton method nonlinear solvers that converge all the time. SPE Journal, 15, 526-544.
Zambra, C., Dumbser, M., Toro, E., & Moraga, N. (2012). A novel numerical method of high-order accuracy for flow in unsaturated porous media. International Journal for Numerical Methods in Engineering, 89, 227-240.
