April  2022, 9(2): 69-84. doi: 10.3934/jcd.2022001

A quadrature-based scheme for numerical solutions to Kirchhoff transformed Richards' equation

1. 

Istituto di Ricerca sulle Acque, Consiglio Nazionale delle Ricerche, Via F. De Blasio 5, 70132 Bari, Italy

2. 

Dipartimento di Matematica, Università degli Studi di Bari Aldo Moro, Via E. Orabona 4, 70125 Bari, Italy

* Corresponding author: Fabio V. Difonzo

Received  February 2021 Revised  January 2022 Published  April 2022 Early access  February 2022

In this work we propose a new numerical scheme for solving Richards' equation within Gardner's framework and accomplishing mass conservation. In order to do so, we resort to Kirchhoff transformation of Richards' equation in mixed form, so to exploit specific Gardner model features, obtaining a linear second order partial differential equation. Then, leveraging the mass balance condition, we integrate both sides of the equation over a generic grid cell and discretize integrals using trapezoidal rule. This approach provides a linear non-homogeneous initial value problem with respect to the Kirchhoff transform variable, whose solution yields the sought numerical scheme. Such a scheme is proven to be $ l^{2} $-stable and convergent to the exact solution under suitably conditions on step-sizes, retaining the order of convergence from the underlying quadrature formula.

Citation: Marco Berardi, Fabio V. Difonzo. A quadrature-based scheme for numerical solutions to Kirchhoff transformed Richards' equation. Journal of Computational Dynamics, 2022, 9 (2) : 69-84. doi: 10.3934/jcd.2022001
References:
[1]

E. AbreuW. LambertJ. Perez and A. Santo, A new finite volume approach for transport models and related applications with balancing source terms, Math. Comput. Simulation, 137 (2017), 2-28.  doi: 10.1016/j.matcom.2016.12.012.

[2]

G. Albuja and A. I. Ávila, A family of new globally convergent linearization schemes for solving Richards' equation, Appl. Numer. Math., 159 (2021), 281-296.  doi: 10.1016/j.apnum.2020.09.012.

[3]

C. AricòM. Sinagra and T. Tucciarelli, The MAST-edge centred lumped scheme for the flow simulation in variably saturated heterogeneous porous media, J. Comput. Phys., 231 (2012), 1387-1425.  doi: 10.1016/j.jcp.2011.10.012.

[4]

L. Beirão da Veiga, A. Pichler and G. Vacca, A virtual element method for the miscible displacement of incompressible fluids in porous media, Comput. Methods Appl. Mech. Engrg., 375 (2021), 35pp. doi: 10.1016/j.cma.2020.113649.

[5]

B. Belfort and F. Lehmann, Comparison of equivalent conductivities for numerical simulation of one-dimensional unsaturated flow, Vadose Zone Journal, 4 (2005), 1191-1200.  doi: 10.2136/vzj2005.0007.

[6]

M. Berardi, Rosenbrock-type methods applied to discontinuous differential systems, Math. Comput. Simulation, 95 (2014), 229-243.  doi: 10.1016/j.matcom.2013.05.006.

[7]

M. BerardiA. AndrisaniL. Lopez and M. Vurro, A new data assimilation technique based on ensemble Kalman filter and Brownian bridges: An application to Richards' equation, Comput. Phys. Comm., 208 (2016), 43-53.  doi: 10.1016/j.cpc.2016.07.025.

[8]

M. BerardiM. D'AbbiccoG. Girardi and M. Vurro, Optimizing water consumption in Richards' equation framework with step-wise root water uptake: A simplified model, Transport in Porous Media, (2022).  doi: 10.1007/s11242-021-01730-y.

[9]

M. Berardi and F. V. Difonzo, Strong solutions for Richards' equation with Cauchy conditions and constant pressure gradient, Environ. Fluid Mech., 20 (2019), 165-174.  doi: 10.1007/s10652-019-09705-w.

[10]

M. BerardiF. V. Difonzo and L. A. Lopez, A mixed MoL-TMoL for the numerical solution of the 2D Richards' equation in layered soils, Comput. Math. Appl., 79 (2020), 1990-2001.  doi: 10.1016/j.camwa.2019.07.026.

[11]

M. BerardiF. V. DifonzoF. Notarnicola and M. Vurro, A transversal method of lines for the numerical modeling of vertical infiltration into the vadose zone, Appl. Numer. Math., 135 (2019), 264-275.  doi: 10.1016/j.apnum.2018.08.013.

[12]

M. BerardiF. V. DifonzoM. Vurro and L. Lopez, The 1D Richards' equation in two layered soils: A Filippov approach to treat discontinuities, Adv. Water Res., 115 (2018), 264-272.  doi: 10.1016/j.advwatres.2017.09.027.

[13]

L. Bergamaschi and M. Putti, Mixed finite elements and Newton-type linearizations for the solution of Richards'equation, Internat. J. Numer. Methods Engrg., 45 (1999), 1025-1046.  doi: 10.1002/(SICI)1097-0207(19990720)45:8<1025::AID-NME615>3.0.CO;2-G.

[14]

P. BroadbridgeE. Daly and J. Goard, Exact solutions of the Richards equation with nonlinear plant-root extraction, Water Resources Res., 53 (2017), 9679-9691.  doi: 10.1002/2017WR021097.

[15]

M. CamporeseE. Daly and C. Paniconi, Catchment-scale Richards equation-based modeling of evapotranspiration via boundary condition switching and root water uptake schemes, Water Resources Res., 51 (2015), 5756-5771.  doi: 10.1002/2015WR017139.

[16]

A. Carminati, A model of root water uptake coupled with rhizosphere dynamics, Vadose Zone Journal, 11 (2012).  doi: 10.2136/vzj2011.0106.

[17]

V. Casulli, A coupled surface-subsurface model for hydrostatic flows under saturated and variably saturated conditions, Internat. J. Numer. Methods Fluids, 85 (2017), 449-464.  doi: 10.1002/fld.4389.

[18]

V. Casulli and P. Zanolli, A nested Newton-type algorithm for finite volume methods solving Richards' equation in mixed form, SIAM J. Sci. Comput., 32 (2010), 2255-2273.  doi: 10.1137/100786320.

[19]

M. A. CeliaE. T. Bouloutas and R. L. Zarba, A general mass-conservative numerical solution for the unsaturated flow equation, Water Resources Res., 26 (1990), 1483-1496.  doi: 10.1029/WR026i007p01483.

[20]

G. M. CocliteA. FanizziL. LopezF. Maddalena and S. F. Pellegrino, Numerical methods for the nonlocal wave equation of the peridynamics, Appl. Numer. Math., 155 (2020), 119-139.  doi: 10.1016/j.apnum.2018.11.007.

[21]

A. ColomboN. Del BuonoL. Lopez and A. Pugliese, Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2911-2934.  doi: 10.3934/dcdsb.2018166.

[22]

A. CoppolaN. ChaaliG. DragonettiN. Lamaddalena and A. Comegna, Root uptake under non-uniform root-zone salinity, Ecohydrology, 8 (2014), 1363-1379.  doi: 10.1002/eco.1594.

[23]

F. Dassi and G. Vacca, Bricks for the mixed high-order virtual element method: Projectors and differential operators, Appl. Numer. Math., 155 (2020), 140-159.  doi: 10.1016/j.apnum.2019.03.014.

[24]

N. Del Buono and L. Lopez, Direct event location techniques based on Adams multistep methods for discontinuous ODEs, Appl. Math. Lett., 49 (2015), 152-158.  doi: 10.1016/j.aml.2015.05.012.

[25]

F. V. DifonzoC. MasciopintoM. Vurro and M. Berardi, Shooting the numerical solution of moisture flow equation with root uptake: A Python tool, Water Res. Mgmt., 35 (2021), 2553-2567.  doi: 10.1007/s11269-021-02850-2.

[26]

M. W. FarthingC. E. Kees and C. T. Miller, Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow, Adv. Water Res., 26 (2003), 373-394.  doi: 10.1016/S0309-1708(02)00187-2.

[27]

M. W. Farthing and F. L. Ogden, Numerical solution of Richards' equation: A review of advances and challenges, Soil Sci. Soc. Amer. J., 81 (2017), 1257-1269.  doi: 10.2136/sssaj2017.02.0058.

[28]

T. P. A. Ferré and A. W. Warrick, Hydrodynamics in soils, in Encyclopedia of Soils in the Environment, Elsevier, Oxford, 2005,227–230. doi: 10.1016/B0-12-348530-4/00377-5.

[29]

B. H. Gilding, Qualitative mathematical analysis of the Richards equation, Transport in Porous Media, 6 (1991), 651-666.  doi: 10.1007/BF00137854.

[30]

M. IcardiP. AsinariD. L. MarchisioS. Izquierdo and R. O. Fox, Quadrature-based moment closures for non-equilibrium flows: Hard-sphere collisions and approach to equilibrium, J. Comput. Phys., 231 (2012), 7431-7449.  doi: 10.1016/j.jcp.2012.07.012.

[31]

K. Kumar, F. List, I. S. Pop and F. A. Radu, Formal upscaling and numerical validation of unsaturated flow models in fractured porous media, J. Comput. Phys., 407 (2020), 21pp. doi: 10.1016/j.jcp.2019.109138.

[32]

W. Lai and F. L. Ogden, A mass-conservative finite volume predictor–corrector solution of the 1D Richards' equation, J. Hydrology, 523 (2015), 119-127.  doi: 10.1016/j.jhydrol.2015.01.053.

[33]

K. Y. LiR. De JongM. T. Coe and N. Ramankutty, Root-water-uptake based upon a new water stress reduction and an asymptotic root distribution function, Earth Interactions, 10 (2006), 1-22.  doi: 10.1175/EI177.1.

[34]

N. LiX. Yue and L. Ren, Numerical homogenization of the Richards equation for unsaturated water flow through heterogeneous soils, Water Resources Res., 52 (2016), 8500-8525.  doi: 10.1002/2015WR018508.

[35]

Z. LiI. Özgen-Xian and F. Z. Maina, A mass-conservative predictor-corrector solution to the 1D Richards equation with adaptive time control, J. Hydrology, 592 (2021).  doi: 10.1016/j.jhydrol.2020.125809.

[36]

Y. LiuH. YangZ. XieP. Qin and R. Li, Parallel simulation of variably saturated soil water flows by fully implicit domain decomposition methods, J. Hydrology, 582 (2020).  doi: 10.1016/j.jhydrol.2019.124481.

[37]

L. Lopez and S. F. Pellegrino, A space-time discretization of a nonlinear peridynamic model on a 2D lamina, Comput. Math. Appl., (2021).  doi: 10.1016/j.camwa.2021.07.004.

[38]

L. Lopez and S. F. Pellegrino, A spectral method with volume penalization for a nonlinear peridynamic model, Internat. J. Numer. Methods Engrg., 122 (2021), 707-725.  doi: 10.1002/nme.6555.

[39]

G. Manzini and S. Ferraris, Mass-conservative finite volume methods on 2-D unstructured grids for the Richards' equation, Adv. Water Res., 27 (2004), 1199-1215.  doi: 10.1016/j.advwatres.2004.08.008.

[40]

W. Merz and P. Rybka, Strong solutions to the Richards equation in the unsaturated zone, J. Math. Anal. Appl., 371 (2010), 741-749.  doi: 10.1016/j.jmaa.2010.05.066.

[41]

P. C. D. Milly, A mass-conservative procedure for time-stepping in models of unsaturated flow, Advances in Water Resources, 8 (1985), 32-36.  doi: 10.1016/0309-1708(85)90078-8.

[42]

K. Mitra and I. S. Pop, A modified L-scheme to solve nonlinear diffusion problems, Comput. Math. Appl., 77 (2019), 1722-1738.  doi: 10.1016/j.camwa.2018.09.042.

[43]

S. M. NaghedifarA. N. Ziaei and S. A. Naghedifar, Optimization of quadrilateral infiltration trench using numerical modeling and Taguchi approach, J. Hydrologic Engrg., 24 (2019).  doi: 10.1061/(ASCE)HE.1943-5584.0001761.

[44] National Research Council, Basic Research Opportunities in Earth Science, The National Academies Press, Washington, DC, 2001.  doi: 10.17226/9981.
[45]

I. NeuweilerD. Erdal and M. Dentz, A non-local Richards equation to model unsaturated flow in highly heterogeneous media under nonequilibrium pressure conditions, Vadose Zone J., 11 (2012).  doi: 10.2136/vzj2011.0132.

[46]

C. Paniconi and M. Putti, Physically based modeling in catchment hydrology at 50: Survey and outlook, Water Resources Res., 51 (2015), 7090-7129.  doi: 10.1002/2015WR017780.

[47]

D. F. RuckerA. W. Warrick and T. P. A. Ferré, Parameter equivalence for the Gardner and van Genuchten soil hydraulic conductivity functions for steady vertical flow with inclusions, Adv. Water Res., 28 (2005), 689-699.  doi: 10.1016/j.advwatres.2005.01.004.

[48]

D. SeusK. MitraI. S. PopF. A. Radu and C. Rohde, A linear domain decomposition method for partially saturated flow in porous media, Comput. Methods Appl. Mech. Engrg., 333 (2018), 331-355.  doi: 10.1016/j.cma.2018.01.029.

[49]

H. Suk and E. Park, Numerical solution of the Kirchhoff-transformed Richards equation for simulating variably saturated flow in heterogeneous layered porous media, J. Hydrology, 579 (2019).  doi: 10.1016/j.jhydrol.2019.124213.

[50]

M. D. TocciC. T. Kelley and C. T. Miller, Accurate and economical solution of the pressure-head form of Richards' equation by the method of lines, Adv. Water Res., 20 (1997), 1-14.  doi: 10.1016/S0309-1708(96)00008-5.

show all references

References:
[1]

E. AbreuW. LambertJ. Perez and A. Santo, A new finite volume approach for transport models and related applications with balancing source terms, Math. Comput. Simulation, 137 (2017), 2-28.  doi: 10.1016/j.matcom.2016.12.012.

[2]

G. Albuja and A. I. Ávila, A family of new globally convergent linearization schemes for solving Richards' equation, Appl. Numer. Math., 159 (2021), 281-296.  doi: 10.1016/j.apnum.2020.09.012.

[3]

C. AricòM. Sinagra and T. Tucciarelli, The MAST-edge centred lumped scheme for the flow simulation in variably saturated heterogeneous porous media, J. Comput. Phys., 231 (2012), 1387-1425.  doi: 10.1016/j.jcp.2011.10.012.

[4]

L. Beirão da Veiga, A. Pichler and G. Vacca, A virtual element method for the miscible displacement of incompressible fluids in porous media, Comput. Methods Appl. Mech. Engrg., 375 (2021), 35pp. doi: 10.1016/j.cma.2020.113649.

[5]

B. Belfort and F. Lehmann, Comparison of equivalent conductivities for numerical simulation of one-dimensional unsaturated flow, Vadose Zone Journal, 4 (2005), 1191-1200.  doi: 10.2136/vzj2005.0007.

[6]

M. Berardi, Rosenbrock-type methods applied to discontinuous differential systems, Math. Comput. Simulation, 95 (2014), 229-243.  doi: 10.1016/j.matcom.2013.05.006.

[7]

M. BerardiA. AndrisaniL. Lopez and M. Vurro, A new data assimilation technique based on ensemble Kalman filter and Brownian bridges: An application to Richards' equation, Comput. Phys. Comm., 208 (2016), 43-53.  doi: 10.1016/j.cpc.2016.07.025.

[8]

M. BerardiM. D'AbbiccoG. Girardi and M. Vurro, Optimizing water consumption in Richards' equation framework with step-wise root water uptake: A simplified model, Transport in Porous Media, (2022).  doi: 10.1007/s11242-021-01730-y.

[9]

M. Berardi and F. V. Difonzo, Strong solutions for Richards' equation with Cauchy conditions and constant pressure gradient, Environ. Fluid Mech., 20 (2019), 165-174.  doi: 10.1007/s10652-019-09705-w.

[10]

M. BerardiF. V. Difonzo and L. A. Lopez, A mixed MoL-TMoL for the numerical solution of the 2D Richards' equation in layered soils, Comput. Math. Appl., 79 (2020), 1990-2001.  doi: 10.1016/j.camwa.2019.07.026.

[11]

M. BerardiF. V. DifonzoF. Notarnicola and M. Vurro, A transversal method of lines for the numerical modeling of vertical infiltration into the vadose zone, Appl. Numer. Math., 135 (2019), 264-275.  doi: 10.1016/j.apnum.2018.08.013.

[12]

M. BerardiF. V. DifonzoM. Vurro and L. Lopez, The 1D Richards' equation in two layered soils: A Filippov approach to treat discontinuities, Adv. Water Res., 115 (2018), 264-272.  doi: 10.1016/j.advwatres.2017.09.027.

[13]

L. Bergamaschi and M. Putti, Mixed finite elements and Newton-type linearizations for the solution of Richards'equation, Internat. J. Numer. Methods Engrg., 45 (1999), 1025-1046.  doi: 10.1002/(SICI)1097-0207(19990720)45:8<1025::AID-NME615>3.0.CO;2-G.

[14]

P. BroadbridgeE. Daly and J. Goard, Exact solutions of the Richards equation with nonlinear plant-root extraction, Water Resources Res., 53 (2017), 9679-9691.  doi: 10.1002/2017WR021097.

[15]

M. CamporeseE. Daly and C. Paniconi, Catchment-scale Richards equation-based modeling of evapotranspiration via boundary condition switching and root water uptake schemes, Water Resources Res., 51 (2015), 5756-5771.  doi: 10.1002/2015WR017139.

[16]

A. Carminati, A model of root water uptake coupled with rhizosphere dynamics, Vadose Zone Journal, 11 (2012).  doi: 10.2136/vzj2011.0106.

[17]

V. Casulli, A coupled surface-subsurface model for hydrostatic flows under saturated and variably saturated conditions, Internat. J. Numer. Methods Fluids, 85 (2017), 449-464.  doi: 10.1002/fld.4389.

[18]

V. Casulli and P. Zanolli, A nested Newton-type algorithm for finite volume methods solving Richards' equation in mixed form, SIAM J. Sci. Comput., 32 (2010), 2255-2273.  doi: 10.1137/100786320.

[19]

M. A. CeliaE. T. Bouloutas and R. L. Zarba, A general mass-conservative numerical solution for the unsaturated flow equation, Water Resources Res., 26 (1990), 1483-1496.  doi: 10.1029/WR026i007p01483.

[20]

G. M. CocliteA. FanizziL. LopezF. Maddalena and S. F. Pellegrino, Numerical methods for the nonlocal wave equation of the peridynamics, Appl. Numer. Math., 155 (2020), 119-139.  doi: 10.1016/j.apnum.2018.11.007.

[21]

A. ColomboN. Del BuonoL. Lopez and A. Pugliese, Computational techniques to locate crossing/sliding regions and their sets of attraction in non-smooth dynamical systems, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 2911-2934.  doi: 10.3934/dcdsb.2018166.

[22]

A. CoppolaN. ChaaliG. DragonettiN. Lamaddalena and A. Comegna, Root uptake under non-uniform root-zone salinity, Ecohydrology, 8 (2014), 1363-1379.  doi: 10.1002/eco.1594.

[23]

F. Dassi and G. Vacca, Bricks for the mixed high-order virtual element method: Projectors and differential operators, Appl. Numer. Math., 155 (2020), 140-159.  doi: 10.1016/j.apnum.2019.03.014.

[24]

N. Del Buono and L. Lopez, Direct event location techniques based on Adams multistep methods for discontinuous ODEs, Appl. Math. Lett., 49 (2015), 152-158.  doi: 10.1016/j.aml.2015.05.012.

[25]

F. V. DifonzoC. MasciopintoM. Vurro and M. Berardi, Shooting the numerical solution of moisture flow equation with root uptake: A Python tool, Water Res. Mgmt., 35 (2021), 2553-2567.  doi: 10.1007/s11269-021-02850-2.

[26]

M. W. FarthingC. E. Kees and C. T. Miller, Mixed finite element methods and higher order temporal approximations for variably saturated groundwater flow, Adv. Water Res., 26 (2003), 373-394.  doi: 10.1016/S0309-1708(02)00187-2.

[27]

M. W. Farthing and F. L. Ogden, Numerical solution of Richards' equation: A review of advances and challenges, Soil Sci. Soc. Amer. J., 81 (2017), 1257-1269.  doi: 10.2136/sssaj2017.02.0058.

[28]

T. P. A. Ferré and A. W. Warrick, Hydrodynamics in soils, in Encyclopedia of Soils in the Environment, Elsevier, Oxford, 2005,227–230. doi: 10.1016/B0-12-348530-4/00377-5.

[29]

B. H. Gilding, Qualitative mathematical analysis of the Richards equation, Transport in Porous Media, 6 (1991), 651-666.  doi: 10.1007/BF00137854.

[30]

M. IcardiP. AsinariD. L. MarchisioS. Izquierdo and R. O. Fox, Quadrature-based moment closures for non-equilibrium flows: Hard-sphere collisions and approach to equilibrium, J. Comput. Phys., 231 (2012), 7431-7449.  doi: 10.1016/j.jcp.2012.07.012.

[31]

K. Kumar, F. List, I. S. Pop and F. A. Radu, Formal upscaling and numerical validation of unsaturated flow models in fractured porous media, J. Comput. Phys., 407 (2020), 21pp. doi: 10.1016/j.jcp.2019.109138.

[32]

W. Lai and F. L. Ogden, A mass-conservative finite volume predictor–corrector solution of the 1D Richards' equation, J. Hydrology, 523 (2015), 119-127.  doi: 10.1016/j.jhydrol.2015.01.053.

[33]

K. Y. LiR. De JongM. T. Coe and N. Ramankutty, Root-water-uptake based upon a new water stress reduction and an asymptotic root distribution function, Earth Interactions, 10 (2006), 1-22.  doi: 10.1175/EI177.1.

[34]

N. LiX. Yue and L. Ren, Numerical homogenization of the Richards equation for unsaturated water flow through heterogeneous soils, Water Resources Res., 52 (2016), 8500-8525.  doi: 10.1002/2015WR018508.

[35]

Z. LiI. Özgen-Xian and F. Z. Maina, A mass-conservative predictor-corrector solution to the 1D Richards equation with adaptive time control, J. Hydrology, 592 (2021).  doi: 10.1016/j.jhydrol.2020.125809.

[36]

Y. LiuH. YangZ. XieP. Qin and R. Li, Parallel simulation of variably saturated soil water flows by fully implicit domain decomposition methods, J. Hydrology, 582 (2020).  doi: 10.1016/j.jhydrol.2019.124481.

[37]

L. Lopez and S. F. Pellegrino, A space-time discretization of a nonlinear peridynamic model on a 2D lamina, Comput. Math. Appl., (2021).  doi: 10.1016/j.camwa.2021.07.004.

[38]

L. Lopez and S. F. Pellegrino, A spectral method with volume penalization for a nonlinear peridynamic model, Internat. J. Numer. Methods Engrg., 122 (2021), 707-725.  doi: 10.1002/nme.6555.

[39]

G. Manzini and S. Ferraris, Mass-conservative finite volume methods on 2-D unstructured grids for the Richards' equation, Adv. Water Res., 27 (2004), 1199-1215.  doi: 10.1016/j.advwatres.2004.08.008.

[40]

W. Merz and P. Rybka, Strong solutions to the Richards equation in the unsaturated zone, J. Math. Anal. Appl., 371 (2010), 741-749.  doi: 10.1016/j.jmaa.2010.05.066.

[41]

P. C. D. Milly, A mass-conservative procedure for time-stepping in models of unsaturated flow, Advances in Water Resources, 8 (1985), 32-36.  doi: 10.1016/0309-1708(85)90078-8.

[42]

K. Mitra and I. S. Pop, A modified L-scheme to solve nonlinear diffusion problems, Comput. Math. Appl., 77 (2019), 1722-1738.  doi: 10.1016/j.camwa.2018.09.042.

[43]

S. M. NaghedifarA. N. Ziaei and S. A. Naghedifar, Optimization of quadrilateral infiltration trench using numerical modeling and Taguchi approach, J. Hydrologic Engrg., 24 (2019).  doi: 10.1061/(ASCE)HE.1943-5584.0001761.

[44] National Research Council, Basic Research Opportunities in Earth Science, The National Academies Press, Washington, DC, 2001.  doi: 10.17226/9981.
[45]

I. NeuweilerD. Erdal and M. Dentz, A non-local Richards equation to model unsaturated flow in highly heterogeneous media under nonequilibrium pressure conditions, Vadose Zone J., 11 (2012).  doi: 10.2136/vzj2011.0132.

[46]

C. Paniconi and M. Putti, Physically based modeling in catchment hydrology at 50: Survey and outlook, Water Resources Res., 51 (2015), 7090-7129.  doi: 10.1002/2015WR017780.

[47]

D. F. RuckerA. W. Warrick and T. P. A. Ferré, Parameter equivalence for the Gardner and van Genuchten soil hydraulic conductivity functions for steady vertical flow with inclusions, Adv. Water Res., 28 (2005), 689-699.  doi: 10.1016/j.advwatres.2005.01.004.

[48]

D. SeusK. MitraI. S. PopF. A. Radu and C. Rohde, A linear domain decomposition method for partially saturated flow in porous media, Comput. Methods Appl. Mech. Engrg., 333 (2018), 331-355.  doi: 10.1016/j.cma.2018.01.029.

[49]

H. Suk and E. Park, Numerical solution of the Kirchhoff-transformed Richards equation for simulating variably saturated flow in heterogeneous layered porous media, J. Hydrology, 579 (2019).  doi: 10.1016/j.jhydrol.2019.124213.

[50]

M. D. TocciC. T. Kelley and C. T. Miller, Accurate and economical solution of the pressure-head form of Richards' equation by the method of lines, Adv. Water Res., 20 (1997), 1-14.  doi: 10.1016/S0309-1708(96)00008-5.

Figure 1.  Output of numerical simulations obtained by MATLAB $ {\mathtt{pdepe}} $ and by the quadrature-based scheme, referred to Example 1
Figure 2.  Output of numerical simulations obtained by MATLAB $ {\mathtt{pdepe}} $ and by quadrature-based scheme, as described in Example 2
Figure 3.  Output of numerical simulations obtained by MATLAB $ {\mathtt{pdepe}} $ and by the quadrature-based scheme, as described in Example 3
Figure 4.  Output of numerical simulations obtained by MATLAB $ {\mathtt{pdepe}} $ and by the quadrature-based scheme (21), as described in Example 4. For the quadrature based scheme we used a constant spatial stepsize of $ \Delta z = 0.0833 $ and a temporal stepsize $ \Delta t = 0.001 $
Table 1.  Numerical orders of convergence of the scheme (21), with $ \Delta z = 3.75\cdot10^{-3}\, $cm and $ \Delta t = 3.125\cdot10^{-4}\, $days, referred to Example 1, providing a mass balance of $ 99.82\% $ according to (32), letting $ \frac{\Delta t}{\Delta z} = 1 $
Step-sizes for $\theta_{\textrm{ref}}$ Numerical order
$ \Delta t, \Delta z $ $ O_{\textrm{num}}^{z}\left(32\Delta t,32\Delta z\right)=1.2054 $
$ O_{\textrm{num}}^{z}\left(16\Delta t,16\Delta z\right)=1.1423 $
$ O_{\textrm{num}}^{z}\left(8\Delta t,8\Delta z\right)=1.1492 $
$ O_{\textrm{num}}^{z}\left(4\Delta t,4\Delta z\right)=1.2508 $
$ O_{\textrm{num}}^{z}\left(2\Delta t,2\Delta z\right)=1.6130 $
Step-sizes for $\theta_{\textrm{ref}}$ Numerical order
$ \Delta t, \Delta z $ $ O_{\textrm{num}}^{z}\left(32\Delta t,32\Delta z\right)=1.2054 $
$ O_{\textrm{num}}^{z}\left(16\Delta t,16\Delta z\right)=1.1423 $
$ O_{\textrm{num}}^{z}\left(8\Delta t,8\Delta z\right)=1.1492 $
$ O_{\textrm{num}}^{z}\left(4\Delta t,4\Delta z\right)=1.2508 $
$ O_{\textrm{num}}^{z}\left(2\Delta t,2\Delta z\right)=1.6130 $
Table 2.  Numerical orders of convergence of the scheme (21), with $ \Delta z = 0.1406 $ cm and $ \Delta t = 0.001 $ days, referred to Example 2, providing a mass balance of $ 100.65\% $ according to (32), letting $ \frac{\Delta t}{\Delta z} = 1 $
Step-sizes for $\theta_{\textrm{ref}}$ Numerical order
$ \Delta t, \Delta z $ $ O_{\textrm{num}}^{z}\left(32\Delta t,32\Delta z\right)=1.4027 $
$ O_{\textrm{num}}^{z}\left(16\Delta t,16\Delta z\right)=1.1968 $
$ O_{\textrm{num}}^{z}\left(8\Delta t,8\Delta z\right)=1.1744 $
$ O_{\textrm{num}}^{z}\left(4\Delta t,4\Delta z\right)=1.2595 $
$ O_{\textrm{num}}^{z}\left(2\Delta t,2\Delta z\right)=1.6032 $
Step-sizes for $\theta_{\textrm{ref}}$ Numerical order
$ \Delta t, \Delta z $ $ O_{\textrm{num}}^{z}\left(32\Delta t,32\Delta z\right)=1.4027 $
$ O_{\textrm{num}}^{z}\left(16\Delta t,16\Delta z\right)=1.1968 $
$ O_{\textrm{num}}^{z}\left(8\Delta t,8\Delta z\right)=1.1744 $
$ O_{\textrm{num}}^{z}\left(4\Delta t,4\Delta z\right)=1.2595 $
$ O_{\textrm{num}}^{z}\left(2\Delta t,2\Delta z\right)=1.6032 $
Table 3.  Numerical orders of convergence of the scheme (21), with $ \Delta z = 0.0026 $ cm and $ \Delta t = 0.0063 $ days, referred to Example 3, providing a mass balance of $ 100.71\% $ according to (32), letting $ \frac{\Delta t}{\Delta z} = 1 $
Step-sizes for $\theta_{\textrm{ref}}$ Numerical order
$ \Delta t, \Delta z $ $ O_{\textrm{num}}^{z}\left(32\Delta t,32\Delta z\right)=3.8094 $
$ O_{\textrm{num}}^{z}\left(16\Delta t,16\Delta z\right)=2.0037 $
$ O_{\textrm{num}}^{z}\left(8\Delta t,8\Delta z\right)=1.0801 $
$ O_{\textrm{num}}^{z}\left(4\Delta t,4\Delta z\right)=1.3516 $
$ O_{\textrm{num}}^{z}\left(2\Delta t,2\Delta z\right)=1.6497 $
Step-sizes for $\theta_{\textrm{ref}}$ Numerical order
$ \Delta t, \Delta z $ $ O_{\textrm{num}}^{z}\left(32\Delta t,32\Delta z\right)=3.8094 $
$ O_{\textrm{num}}^{z}\left(16\Delta t,16\Delta z\right)=2.0037 $
$ O_{\textrm{num}}^{z}\left(8\Delta t,8\Delta z\right)=1.0801 $
$ O_{\textrm{num}}^{z}\left(4\Delta t,4\Delta z\right)=1.3516 $
$ O_{\textrm{num}}^{z}\left(2\Delta t,2\Delta z\right)=1.6497 $
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