Truncation error and stability analysis of iterative and non‐iterative Thomas–Gladwell methods for first‐order non‐linear differential equations

Kavetski, Dmitri; Binning, Philip John; Sloan, Scott W.

doi:10.1002/nme.1035

Cited by 15 publications

(14 citation statements)

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“…However, for these three test cases and, on average, the best performance in terms of efficiency was obtained using a stopping criterion based on truncation error with its corresponding time-step strategy (ST_ ψ). Similar results were obtained by Kavetski et al (2001) for the pressurebased RE and by Kavetski and Binning (2004) for the moisture-based RE.…”

Section: Discussionsupporting

confidence: 85%

“…The adaptive scheme used in this work evaluates the time steps through truncation error due to the temporal discretization as proposed by Thomas and Gladwell (1988). This scheme was already applied to the pressure-based formulation by Kavetski et al (2001) and to the moisture-based formulation by Kavetski and Binning (2004).…”

Section: Algorithms and Time-stepping Strategymentioning

confidence: 99%

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Ross scheme, Newton–Raphson iterative methods and time-stepping strategies for solving the mixed form of Richards' equation

Maina

Ackerer

2017

Hydrol. Earth Syst. Sci.

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Abstract. The solution of the mathematical model for flow in variably saturated porous media described by the Richards equation (RE) is subject to heavy numerical difficulties due to its highly nonlinear properties and remains very challenging. Two different algorithms are used in this work to solve the mixed form of RE: the traditional iterative algorithm and a time-adaptive algorithm consisting of changing the timestep magnitude within the iteration procedure while the nonlinear parameters are computed with the state variable at the previous time. The Ross method is an example of this type of scheme, and we show that it is equivalent to the NewtonRaphson method with a time-adaptive algorithm.Both algorithms are coupled to different time-stepping strategies: the standard heuristic approach based on the number of iterations and two strategies based on the time truncation error or on the change in water saturation. Three different test cases are used to evaluate the efficiency of these algorithms.The numerical results highlight the necessity of implementing an estimate of the time truncation errors.

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Section: Discussionsupporting

confidence: 85%

Section: Algorithms and Time-stepping Strategymentioning

confidence: 99%

Ross scheme, Newton–Raphson iterative methods and time-stepping strategies for solving the mixed form of Richards' equation

Maina

Ackerer

2017

Hydrol. Earth Syst. Sci.

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“…If the stage quantities (20) are inserted into the definition of the stage derivative (21), one arrives at a system of non-linear equations in each stage…”

Section: Time Discretisation Using Rosenbrock Methodsmentioning

confidence: 99%

“…The applicability of further time-adaptive methods, see, for example, [20], are worth being studied in the future. In this context, it has to be emphasised that the DAE-approach incorporates an essential by-product, namely time-adaptivity, which leads both to more accurate results (control of local integration error), and a stable procedure as well.…”

mentioning

confidence: 99%

Finite element analysis of viscoelastic structures using Rosenbrock-type methods

Hartmann

Wensch

2006

Comput Mech

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The consistent application of the space-time discretisation in the case of quasi-static structural problems based on constitutive equations of evolutionary type yields after the spatial discretisation by means of the finite element method a system of differential-algebraic equations. In this case the resulting system of differential-algebraic equations with the unknown nodal displacements and the evolution equations at all spatial quadrature points of the finite element discretisation are solved by means of a time-adaptive Rosenbrock-type methods leading to an iteration-less solution scheme in non-linear finite element analysis. The applicability of the method will be studied by means of a simple example of a viscoelastic structure.

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“…The norm for the numerical solutions of Richards' equation is low-order time discretizations, which are globally first-order. Notable exceptions include the approaches in (Tocci et al, 1997;Farthing et al, 2003b) based on higher order Backward Difference Formulas (BDFs) and the second-order TaylorGladwell scheme introduced in (Kavetski et al, 2001a(Kavetski et al, , 2004. The vast majority of temporal approximations are also implicit in head-based models that allow fully saturated conditions to develop in the domain (Celia et al, 1990).…”

Section: Temporal Discretizationmentioning

confidence: 99%

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

Farthing

Ogden

2017

Soil Science Soc of Amer J

245

138

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The flow of water in partially saturated porous media is of importance in fields such as hydrology, agriculture, environment and waste management. It is also one of the most complex flows in nature. The Richards' equation describes the flow of water in an unsaturated porous medium due to the actions of gravity and capillarity neglecting the flow of the non-wetting phase, usually air. Analytical solutions of Richards' equation exist only for simplified cases, so most practical situations require a numerical solution in one-two-or threedimensions, depending on the problem and complexity of the flow situation. Despite the fact that the first reasonably complete conservative numerical solution method was published in the early 1990s, the numerical solution of the Richards' equation remains computationally expensive and in certain circumstances, unreliable. A universally robust and accurate solution methodology has not yet been identified that is applicable across the range of soils, initial and boundary conditions found in practice. Existing solution codes have been modified over years to attempt to increase robustness. Despite theoretical results on the existence of solutions given sufficiently regular data and constitutive relations, our numerical methods often fail to demonstrate reliable convergence behavior in practice, especially for higher-order methods. Because of robustness, the lack of higher-order accuracy and computational expense, alternative solution approaches or methods are needed. There is also a need for better documentation of improved solution methodologies and benchmark test problems to facilitate consistent advances and avoid re-inventing of the wheel.T his review paper is intended to serve as a touchstone on the state of the science of calculating the flow of water through unsaturated porous media. As active researchers we believe it is good to occasionally take stock in what has been accomplished up to date, where things may be headed, and what important issues remain. This review concentrates on computational modeling of flow through the unsaturated zone via Richards' equation.This effort can help provide some focus to the research community and serve to help propel new research forward, particularly by those just joining the field. There are many practical problems that require calculation of one-, two-, and threedimensional fluxes in the unsaturated zone. For a much broader review of modeling soil processes, we recommend the recent paper from Vereecken et al. (2016). A detailed review of mathematical models of infiltration can be found (Assouline, 2013), while Paniconi and Putti (2015) provide historical perspective and an overview of modeling Richards' equation in the context of catchment hydrology.The equation attributed to Richards (1931) that describes the flow of water through unsaturated porous media under the action of capillarity and gravity was first published by the English mathematician and physicist Lewis Fry Richardson in 1922(Richardson, 1922. Therefore, the equation would r...

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Truncation error and stability analysis of iterative and non‐iterative Thomas–Gladwell methods for first‐order non‐linear differential equations

Cited by 15 publications

References 13 publications

Ross scheme, Newton–Raphson iterative methods and time-stepping strategies for solving the mixed form of Richards' equation

Ross scheme, Newton–Raphson iterative methods and time-stepping strategies for solving the mixed form of Richards' equation

Finite element analysis of viscoelastic structures using Rosenbrock-type methods

Numerical Solution of Richards' Equation: A Review of Advances and Challenges

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