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On the Shooting Method Applied to Richards’ Equation with a Forcing Term

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Computational Science and Its Applications – ICCSA 2021 (ICCSA 2021)

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Abstract

The problem of modeling water flow in the root zone with plant root absorption is of crucial importance in many environmental and agricultural issues, and is still of interest in the applied mathematics community. In this work we propose a formal justification and a theoretical background of a recently introduced numerical approach, based on the shooting method, for integrating the unsaturated flow equation with a sink term accounting for the root water uptake model. Moreover, we provide various numerical simulations for this method, comparing the results with the numerical solutions obtained by MATLAB pdepe .

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Notes

  1. 1.

    We highlight that the uptake function R should be used as reported here, instead as the one originally proposed in [22], in order for (23) to hold; the function D stays the same.

References

  1. Maggi, S.: Estimating water retention characteristic parameters using differential evolution. Comput. Geotech. 86, 163–172 (2017). https://doi.org/10.1016/j.compgeo.2016.12.025

    Article  Google Scholar 

  2. Friedman, S.P., Communar, G., Gamliel, A.: DIDAS - user-friendly software package for assisting drip irrigation design and scheduling. Comput. Electron. Agricult. 120, 36–52 (2016). https://doi.org/10.1016/j.compag.2015.11.007

    Article  Google Scholar 

  3. Brunetti, G., Šimůnek, J., Bautista, E.: A hybrid finite volume-finite element model for the numerical analysis of furrow irrigation and fertigation. Comput. Electron. Agricult. 150, 312–327 (2018). https://doi.org/10.1016/j.compag.2018.05.013

    Article  Google Scholar 

  4. Arbat, G., Puig-Bargués, J., Duran-Ros, M., Barragán, J., de Cartagena, F.R.: Drip-Irriwater: computer software to simulate soil wetting patterns under surface drip irrigation. Comput. Electron. Agricult. 98, 183–192 (2013). https://doi.org/10.1016/j.compag.2013.08.009

    Article  Google Scholar 

  5. Berardi, M., Difonzo, F., Notarnicola, F., Vurro, M.: A transversal method of lines for the numerical modeling of vertical infiltration into the vadose zone. Appl. Numer. Math. 135, 264–275 (2019). https://doi.org/10.1016/j.apnum.2018.08.013

    Article  MathSciNet  MATH  Google Scholar 

  6. Berardi, M., Difonzo, F.V.: Strong solutions for Richards’ equation with Cauchy conditions and constant pressure gradient. Environ. Fluid Mech. 20(1), 165–174 (2019). https://doi.org/10.1007/s10652-019-09705-w

    Article  Google Scholar 

  7. Casulli, V.: A coupled surface-subsurface model for hydrostatic flows under saturated and variably saturated conditions. Int. J. Numer. Meth. Fluids 85(8), 449–464 (2017). https://doi.org/10.1002/fld.4389

    Article  MathSciNet  Google Scholar 

  8. Nordbotten, J.M., Boon, W.M., Fumagalli, A., Keilegavlen, E.: Unified approach to discretization of flow in fractured porous media. Comput. Geosci. 23(2), 225–237 (2018). https://doi.org/10.1007/s10596-018-9778-9

    Article  MathSciNet  MATH  Google Scholar 

  9. Berre, I., et al.: Verification benchmarks for single-phase flow in three-dimensional fractured porous media. Adv. Water Resour. 147, 103759 (2021). https://doi.org/10.1016/j.advwatres.2020.103759

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. Lopez, L., Maset, S.: Time-transformations for the event location in discontinuous ODEs. Math. Comp. 87(3), 2321–2341 (2018). https://doi.org/10.1090/mcom/3305

    Article  MathSciNet  MATH  Google Scholar 

  12. Del Buono, N., Elia, C., Lopez, L.: On the equivalence between the sigmoidal approach and Utkin’s approach for piecewise-linear models of gene regulatory. SIAM J. Appl. Dyn. Syst. 13(3), 1270–1292 (2014). https://doi.org/10.1137/130950483

    Article  MathSciNet  MATH  Google Scholar 

  13. Berardi, M., Difonzo, F., Vurro, M., Lopez, L.: The 1D Richards’ equation in two layered soils: a Filippov approach to treat discontinuities. Adv. Water Resour. 115, 264–272 (2018). https://doi.org/10.1016/j.advwatres.2017.09.027

    Article  Google Scholar 

  14. Berardi, M., Difonzo, F., Lopez, L.: A mixed MoL-TMoL for the numerical solution of the 2D Richards’ equation in layered soils. Comput. Math. Appl. 79(7), 1990–2001 (2019). https://doi.org/10.1016/j.camwa.2019.07.026

    Article  MathSciNet  MATH  Google Scholar 

  15. Difonzo, F.V., Masciopinto, C., Vurro, M., Berardi, M.: Shooting the numerical solution of moisture flow equation with root water uptake models: a Python tool. Water Resour. Manage. 35(8), 2553–2567 (2021). https://doi.org/10.1007/s11269-021-02850-2

    Article  Google Scholar 

  16. Camporese, M., Daly, E., Paniconi, C.: Catchment-scale Richards equation-based modeling of evapotranspiration via boundary condition switching and root water uptake schemes. Water Resour. Res. 51(7), 5756–5771 (2015). https://doi.org/10.1002/2015WR017139

    Article  Google Scholar 

  17. Manoli, G., Bonetti, S., Scudiero, E., Morari, F., Putti, M., Teatini, P.: Modeling soil-plant dynamics: assessing simulation accuracy by comparison with spatially distributed crop yield measurements. Vadose Zone J. 14(2) (2015), vzj2015-05. https://doi.org/10.2136/vzj2015.05.0069

  18. Roose, T., Fowler, A.: A model for water uptake by plant roots. J. Theor. Biol. 228(2), 155–171 (2004). https://doi.org/10.1016/j.jtbi.2003.12.012

    Article  MathSciNet  MATH  Google Scholar 

  19. Couvreur, V., Vanderborght, J., Javaux, M.: A simple three-dimensional macroscopic root water uptake model based on the hydraulic architecture approach. Hydrol. Earth Syst. Sci. 16(8), 2957–2971 (2012). https://doi.org/10.5194/hess-16-2957-2012

    Article  Google Scholar 

  20. Albrieu, J.L.B., Reginato, J.C., Tarzia, D.A.: Modeling water uptake by a root system growing in a fixed soil volume. Appl. Math. Model. 39(12), 3434–3447 (2015). https://doi.org/10.1016/j.apm.2014.11.042

    Article  MathSciNet  MATH  Google Scholar 

  21. Simunek, J., Hopmans, J.W.: Modeling compensated root water and nutrient uptake. Ecol. Model. 220(4), 505–521 (2009). https://doi.org/10.1016/j.ecolmodel.2008.11.004

    Article  Google Scholar 

  22. Broadbridge, P., Daly, E., Goard, J.: Exact solutions of the Richards equation with nonlinear plant-root extraction. Water Resour. Res. 53(11), 9679–9691 (2017). https://doi.org/10.1002/2017WR021097

    Article  Google Scholar 

  23. Mathur, S., Rao, S.: Modeling water uptake by plant roots. J. Irrig. Drain. Eng. 125(3), 159–165 (1999). https://doi.org/10.1061/(ASCE)0733-9437(1999)125:3(159)

    Article  Google Scholar 

  24. D’Abbicco, M., Buono, N.D., Gena, P., Berardi, M., Calamita, G., Lopez, L.: A model for the hepatic glucose metabolism based on hill and step functions. J. Comput. Appl. Math. 292, 746–759 (2016). https://doi.org/10.1016/j.cam.2015.01.036

    Article  MathSciNet  MATH  Google Scholar 

  25. Feddes, R.A., Kowalik, P., Kolinska-Malinka, K., Zaradny, H.: Simulation of field water uptake by plants using a soil water dependent root extraction function. J. Hydrol. 31(1), 13–26 (1976). https://doi.org/10.1016/0022-1694(76)90017-2

    Article  Google Scholar 

  26. Berardi, M., Girardi, G., D’Abbico, M., Vurro, M.: Optimizing water consumption in Richards’ equation framework with step-wise root water uptake: a simplified model (2021, submitted)

    Google Scholar 

  27. Jarvis, N.: A simple empirical model of root water uptake. J. Hydrol. 107(1), 57–72 (1989). https://doi.org/10.1016/0022-1694(89)90050-4

    Article  Google Scholar 

  28. Babazadeh, H., Tabrizi, M., Homaee, M.: Assessing and modifying macroscopic root water extraction basil (ocimum basilicum) models under simultaneous water and salinity stresses. Soil Sci. Soc. Am. J. 81, 10–19 (2017). https://doi.org/10.2136/sssaj2016.07.0217

    Article  Google Scholar 

  29. Coppola, A., Chaali, N., Dragonetti, G., Lamaddalena, N., Comegna, A.: Root uptake under non-uniform root-zone salinity. Ecohydrology 8(7), 1363–1379 (2015). https://doi.org/10.1002/eco.1594

    Article  Google Scholar 

  30. Bergamaschi, L., Putti, M.: Mixed finite elements and Newton-type linearizations for the solution of Richards’ equation. Int. J. Numer. Meth. Eng. 45, 1025–1046 (1999). https://doi.org/10.1002/(SICI)1097-0207(19990720)45:8<1025::AID-NME615>3.0.CO;2-G

  31. Li, N., Yue, X., Ren, L.: Numerical homogenization of the Richards equation for unsaturated water flow through heterogeneous soils. Water Resour. Res. 52(11), 8500–8525 (2016). https://doi.org/10.1002/2015WR018508

    Article  Google Scholar 

  32. Suk, H., Park, E.: Numerical solution of the Kirchhoff-transformed Richards equation for simulating variably saturated flow in heterogeneous layered porous media. J. Hydrol. 579, 124213 (2019). https://doi.org/10.1016/j.jhydrol.2019.124213

    Article  Google Scholar 

  33. Mathews, J.H., Fink, K.K.: Numerical Methods Using MATLAB, 4th edn. Pearson, New York (2004)

    Google Scholar 

  34. D’Ambrosio, R., Jackiewicz, Z.: Construction and implementation of highly stable two-step continuous methods for stiff differential systems. Math. Comput. Simul. 81(9), 1707–1728 (2011). https://doi.org/10.1016/j.matcom.2011.01.005

    Article  MathSciNet  MATH  Google Scholar 

  35. Carsel, R.F., Parrish, R.S.: Developing joint probability distributions of soil water retention characteristics. Water Resour. Res. 24(5), 755–769 (1988). https://doi.org/10.1029/WR024i005p00755

    Article  Google Scholar 

  36. Li, K.Y., De Jong, R., Coe, M.T., Ramankutty, N.: Root-water-uptake based upon a new water stress reduction and an asymptotic root distribution function. Earth Interact. 10(14), 1–22 (2006)

    Article  Google Scholar 

  37. Berardi, M., Difonzo, F.: A quadrature-based scheme for numerical solutions to Kirchhoff transformed Richards’ equation (2021, submitted)

    Google Scholar 

  38. Lopez, L., Pellegrino, S.F.: A spectral method with volume penalization for a nonlinear peridynamic model. Int. J. Numer. Meth. Eng. 122(3), 707–725 (2020). https://doi.org/10.1002/nme.6555

    Article  MathSciNet  Google Scholar 

  39. Lopez, L., Pellegrino, S.F.: A space-time discretization of a nonlinear peridynamic model on a 2D lamina (2021)

    Google Scholar 

  40. Lopez, L., Pellegrino, S.F., Coclite, G., Maddalena, F., Fanizzi, A.: Numerical methods for the nonlocal wave equation of the peridynamics. Appl. Numer. Math. 155, 119–139 (2018). https://doi.org/10.1016/j.apnum.2018.11007

    Article  MathSciNet  MATH  Google Scholar 

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

  42. Kroener, E.: How mucilage affects soil hydraulic dynamics. J. Plant Nutr. Soil Sci. 184, 20–24 (2021). https://doi.org/10.1002/jpln.202000545

    Article  Google Scholar 

  43. Wu, X., Zuo, Q., Shi, J., Wang, L., Xue, X., Ben-Gal, A.: Introducing water stress hysteresis to the Feddes empirical macroscopic root water uptake model. Agricult. Water Manage. 240, 106293 (2020). https://doi.org/10.1016/j.agwat.2020.106293

    Article  Google Scholar 

  44. Tubini, N., Gruber, S., Rigon, R.: A method for solving heat transfer with phase change in ice or soil that allows for large time steps while guaranteeing energy conservation. Cryosphere Discuss. 2020, 1–41 (2020). https://doi.org/10.5194/tc-2020-293

    Article  Google Scholar 

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Acknowledgments

The first author has been funded by REFIN Project, grant number 812E4967. The second author acknowledges the partial support of the project RIUBSAL (grant number 030_DIR_2020_00178), funded by Regione Puglia through the P.S.R. Puglia 2014/2020 - Misura 16 – Cooperazione - Sottomisura 16.2 call.

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Authors and Affiliations

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

    Fabio Vito Difonzo

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

    Giovanni Girardi

Authors
  1. Fabio Vito Difonzo
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Correspondence to Fabio Vito Difonzo .

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Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Potenza, Italy

    Beniamino Murgante

  3. Covenant University, Ota, Nigeria

    Sanjay Misra

  4. University of Cagliari, Cagliari, Italy

    Chiara Garau

  5. University of Cagliari, Cagliari, Italy

    Ivan Blečić

  6. Monash University, Clayton, VIC, Australia

    David Taniar

  7. Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

  8. University of Minho, Braga, Portugal

    Ana Maria A. C. Rocha

  9. Polytechnic University of Bari, Bari, Italy

    Eufemia Tarantino

  10. Polytechnic University of Bari, Bari, Italy

    Carmelo Maria Torre

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Difonzo, F.V., Girardi, G. (2021). On the Shooting Method Applied to Richards’ Equation with a Forcing Term. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2021. ICCSA 2021. Lecture Notes in Computer Science(), vol 12949. Springer, Cham. https://doi.org/10.1007/978-3-030-86653-2_20

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  • DOI: https://doi.org/10.1007/978-3-030-86653-2_20

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