Keywords:-

Keywords: Variably saturated flow; Finite difference; Numerical solution; Richards' equation; Method of lines

Article Content:-

Abstract

Robust, accurate and efficient numerical simulation of groundwater flow in the unsaturated zone remains computationally expensive, particularly for problems that involved sharp fronts in both space and time. Numerical solution of these problems along with standard approaches that use of uniform spatial and temporal discretizations leads to inefficient and expensive simulations. Accurate solution of pressure head form of Richards' equation is very difficult using standard time integration techniques because in the time integration process the mass balance errors grows unless very small time steps are used. Richards' equation may be solved for many problems more economically and robustly with variable time step size instead of constant time step size. We solve Richards' equation using the method of lines with a standard finite difference technique. We show how a differential algebraic equation implementation of the method of lines can give solutions to Richards' equation that are accurate, have good mass balance properties, and are more economical for a wide range of solution accuracy. We implement the method of lines using four higher order time integration MATLAB ODE solvers ode15s, ode23s, ode23t and ode23tb to (i) assure robustness for difficult nonlinear problems and computational efficiency; (ii) develop higher order adaptive methods for the time; (iii) investigate the advantage of using higher-order methods in time; and (iv) compare the computational performance of the ODE solvers. The numerical results demonstrate that the proposed method provides a robust and efficient alternative to standard approaches for simulating variably saturated flow in one spatial dimension.

References:-

References

Williams, G. A. and Miller, C. T., An evaluation of temporally adaptive transformation approaches for solving Richards’ equation,

Adv. Water Resources, 1999;22(8):831-840.

Arampatzis, G et.al., Estimation of unsaturated flow in layered soils with the finite control volume method, Irrigation and Drainage,

o1;5:349-358.

Koorevaar, P. et.al., Elements of soil physics: Developments in Soil Science (Elsevier Science Publishers B.V., The Netherlands,

.

Miller, C. T. et.al., A spatially and temporally adaptive solution of Richards’ equation, Adv. Water Resources, 2005;29:525-545.

Celia, M. A. et.al., A General mass-conservative numerical solution for the unsaturated flow equation, Water Resources Res.,

;26(7):1483-1496.

Tocci, M. D. et. al., Accurate and economical solution of the pressure-head form of Richards’ equation by the method of lines, Adv.

Water Resources, 1997;20(1):1-14.

Hanks, R. J. and Bowers, S. A., Numerical solution of the moisture flow equation for the infiltration into layered soils, Soil. Sci. Proc.,

:530-534.

Romano, N. et.al., Numerical analysis of one dimensional unsaturated flow in layered soils, Adv. Water Resources, 1998;21:315-324.

Abriola, L. M. and Lang, J. R., Self-adaptive finite element solution of the one dimensional unsaturated flow equation, Int. J. Numer.

Methods Fluids, 1990;10: 227-246.

Ju, S. H. and Kung, K. J. S., Mass types, Element orders and Solution schemes for Richards’ equation, Computers and Geosciences,

;23(2):175-187.

Milly, P. C. D., A mass-conservative procedure for time-stepping in models of unsaturated flow, Adv. Water Resources, 1985;8:32-36.

Kavetski, D., et.al., Noniterative time stepping schemes with adaptive truncation error control for the solution of Richards’ equation,

Water Resources Res., 2002;38(10):1211-1220.

Kavetski, D., et.al., Adaptive backward Euler time stepping with truncation error control for numerical modelling of unsaturated fluid

flow, Int. J. Numer. Meth. Eng., 2001a ;53: 1301-1322.

Fassino, C. and Manzini, G., Fast-secant algorithms for the non-linear Richards Equation. Communications in Numerical Methods in

engineering, 1998;14: 921-930.

Bergamaschi, L. and Putti, M., Mixed finite elements and Newton-type linearizations for the solution of Richards’ equation, Int. J.

Numer. Meth. Eng., 1999;45:1025-1046.

Lehmann, F. and Ackerer, P. H., Comparison of iterative methods for improved solutions for fluid flow equation in partially saturated

porous media, Transport in Porous Media, 1998;31:275-292.

Saucez, P. et.al., An adaptive method of lines solution of the Korteweg-de Vries equation, Comput. Math. Appl., 1998;35(12):13–25.

Schiesser, W. E., Method of lines solution of the Korteweg-de Vries equation, Comput. Math. Appl., 1994;28(10–12):147–54.

Brenan, K. E., et.al., The numerical solution of initial value problems in differential–algebraic equations (Philadelphia, PA: Soc. Ind.

Appl. Math., 1996).

Kees, C. E. and Miller, C. T., C++ implementations of numerical methods for solving differential–algebraic equations: design and

optimization considerations, ACM Trans Math Software, 1999;25(4): 377–403.

Miller, C. T., et.al., Robust solution of Richards' equation for non uniform porous media, Water Resources Res., 1998;34:2599-2610.Tocci, M. D., et.al., Inexact Newton methods and the method of lines for solving Richards’ equation in two space dimensions, Comput.

Geosci., 1999;2(4); 291–309.

Farthing, M. W., et.al., Mixed finite element methods and higher-order temporal approximations, Adv. Water Resources, 2002;25(1);

–101.

Farthing, M. W., et.al., Mixed finite element methods and higher order temporal approximations for variably saturated groundwater

flow, Adv Water Resources, 2003;26(4);373–94.

Mansell, R. S., et.al., Adaptive Grid Refinement in Numerical Models for Water Flow and Chemical Transport in Soil: A Review,

Vadose Zone Journal, 2002;1:222-238.

Kavetski, D., et.al., Adaptive time stepping and error control in a mass conservative numerical solution of the mixed form of Richards

equation, Adv. Water Resources, 2001b;24:595-605.

Baca, R. G., et.al., Mixed transform finite element method for solving the non-linear equation for flow in variably saturated porous

media, Int. J. Numer. Meth. Fluids, 1997;24:441-455.

Guarracino, L. and Quintana, F., A third-order accurate time scheme for variably saturated groundwater flow modeling,

Communications in Numerical Methods in engineering, 2004;20:379-389.

van Genuchten, M. T., A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils, Soil Sci. Soc. Am. J.,

;44:892–898.

Rathfelder, K. and Abriola, L. M., Mass conservative numerical solutions of the head-based Richards equation, Water Resources Res.,

;30(9):2579-2586.

Schiesser, W. E., The numerical method of lines: Integration of Partial Differential equations: ODEs, DAEs and PDEs (Academic

Press: San Diego, 1991).

Shampine, L. F. and Reichelt, M. W., The MATLAB ODE Suite Report 94-6 (Math. Dept. SMU, Dallas, 1994).

Shampine, L. F. and Reichelt, M. W., The MATLAB ODE Suit, SIAM J. Sci. Comput., 1997;18:1-22.

Shampine, L. F., et.al., Solving Index-1 DAEs in MATLAB and Simulink, SIAM Review, 1999;41:538-552.

Shampine, L. F., Numerical Solution of ordinary Differential Equations (Chapman and Hall, Newyork, 1994).

Loague, K. and Green, R. E., Statistical and graphical methods for evaluating solute transport models: Overview and application, J.

Contam. Hydrol., 1991;7:51–73.

Kool, J. B. and Parker, J. C., Development and evaluation of closed form expressions for hysteretic soil hydraulic properties, Water

Resources Res., 1987;23(1):105-114.

Kees, C. E. and Miller, C. T., Higher order time integration methods for two-phase flow, Adv. Water Resources, 2002;25(2):159–77.

Downloads

Citation Tools

How to Cite
Islam, M. S., & Hasan, M. K. (2014). Accurate and economical solution of Richards’ equation by the method of lines and comparison of the computational performance of ODE solvers. International Journal Of Mathematics And Computer Research, 2(02), 328-346. Retrieved from http://ijmcr.in/index.php/ijmcr/article/view/105