Asymptotical computations for a model of flow in saturated porous media

Amodio, Pierluigi; Budd, C. J.; Koch, Othmar; Settanni, Giuseppina; Weinmüller, Ewa

doi:10.1016/j.amc.2014.03.063

Cited by 9 publications

(18 citation statements)

References 15 publications

(39 reference statements)

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“…Neglecting the

O (γ^{- 4})

terms yields a quadratic equation for γ 2 whose solution can be expressed for large

\overset{β}{true}

γ = {\overset{β}{true}}^{1 / 2} - \frac{1}{4} {\overset{β}{true}}^{- 1 / 2} - (\frac{α_{3}}{2} + \frac{1}{32}) {\overset{β}{true}}^{- 3 / 2} + O ({\bar{β}}^{- 5 / 2}) .

The high‐fidelity numerical calculations reported by Amodio et al. provide convincing numerical evidence that

α_{3} = \frac{1}{12},

although we have no formal calculation as yet to confirm this result.…”

Section: Multilayer Asymptotic Solutionmentioning

confidence: 73%

“…With this restricted range of γ, we can still test the validity of the asymptotic formulas derived here but we are not able to see their true accuracy at larger γ, as the asymptotic series we have derived only converge at a polynomial rate as γ increases. Consequently, we will also report some results using a much more sophisticated numerical approach that is the subject of a related paper and which employs a high‐order Taylor‐expansion‐based boundary value solver coupled with mesh adaptivity. This method permits calculations up to

γ = 18

, corresponding to

\bar{β} \approx 325

and

θ_{\infty} \approx 1.18 \times 10^{- 141}

.…”

Section: Comparison Of Asymptotic and Numerical Resultsmentioning

confidence: 99%

“…Although our simple numerical approach fails for

\bar{β} > 20

, a more sophisticated numerical algorithm has been implemented by Amodio et al. that yields accurate solutions for values of

\overset{β}{true}

much larger. Their results for a much wider range of

\overset{β}{true}

are summarized in Table , along with the corresponding asymptotic results for

y_{*}^{asy}

y_{*}^{B}

, and

y_{*}^{P}

.…”

Section: Comparison Of Asymptotic and Numerical Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Multilayer Asymptotic Solution for Wetting Fronts in Porous Media with Exponential Moisture Diffusivity

Budd

2015

Stud Appl Math

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We study the asymptotic behaviour of sharp front solutions arising from the nonlinear diffusion equation θ t = (D(θ)θ x ) x , where the diffusivity is an exponential function D(θ) = D o exp(βθ). This problem arises for example in the study of unsaturated flow in porous media where θ represents the liquid saturation. For physical parameters corresponding to actual porous media, the diffusivity at the residual saturation is D(0) = D o ≪ 1 so that the diffusion problem is nearly degenerate. Such problems are characterised by wetting fronts that sharply delineate regions of saturated and unsaturated flow, and that propagate with a well-defined speed. Using matched asymptotic expansions in the limit of large β, we derive an analytical description of the solution that is uniformly valid throughout the wetting front. This is in contrast with most other related analyses that instead truncate the solution at some specific wetting front location, which is then calculated as part of the solution, and beyond that location the solution is undefined. Our asymptotic analysis demonstrates that the solution has a four-layer structure, and by matching through the adjacent layers we obtain an estimate of the wetting front location in terms of the material parameters describing the porous medium. Using numerical simulations of the original nonlinear diffusion equation, we demonstrate that the first few terms in our series solution provide approximations of physical quantities such as wetting front location and speed of propagation that are more accurate (over a wide range of admissible β values) than other asymptotic approximations reported in the literature.

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“…Neglecting the

O (γ^{- 4})

terms yields a quadratic equation for γ 2 whose solution can be expressed for large

\overset{β}{true}

γ = {\overset{β}{true}}^{1 / 2} - \frac{1}{4} {\overset{β}{true}}^{- 1 / 2} - (\frac{α_{3}}{2} + \frac{1}{32}) {\overset{β}{true}}^{- 3 / 2} + O ({\bar{β}}^{- 5 / 2}) .

The high‐fidelity numerical calculations reported by Amodio et al. provide convincing numerical evidence that

α_{3} = \frac{1}{12},

although we have no formal calculation as yet to confirm this result.…”

Section: Multilayer Asymptotic Solutionmentioning

confidence: 73%

γ = 18

, corresponding to

\bar{β} \approx 325

and

θ_{\infty} \approx 1.18 \times 10^{- 141}

.…”

Section: Comparison Of Asymptotic and Numerical Resultsmentioning

confidence: 99%

“…Although our simple numerical approach fails for

\bar{β} > 20

, a more sophisticated numerical algorithm has been implemented by Amodio et al. that yields accurate solutions for values of

\overset{β}{true}

much larger. Their results for a much wider range of

\overset{β}{true}

are summarized in Table , along with the corresponding asymptotic results for

y_{*}^{asy}

y_{*}^{B}

, and

y_{*}^{P}

.…”

Section: Comparison Of Asymptotic and Numerical Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Multilayer Asymptotic Solution for Wetting Fronts in Porous Media with Exponential Moisture Diffusivity

Budd

2015

Stud Appl Math

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show abstract

“…We mention Initial Value Problems [4], Boundary Value Problems [5,6,11,12] and Sturm-Liouville problems [7][8][9][10]. In all these cases the proposed formulae show good stability properties and the developed codes turn out to be competitive with the existing software, in particular when the problem to be solved is stiff and the solution has a slope different from that of its derivatives.…”

Section: High Order Finite Difference Schemesmentioning

confidence: 99%

“…. ; n À 1, respectively, andṽ contains the coefficients of the quadrature formula discretizing (4). We remark that the order of this last formula does not entail a reduction of the order of the overall method since it represents a scaling factor used to fix a particular eigenfunction.…”

Section: High Order Finite Difference Schemesmentioning

confidence: 99%

Reprint of Variable-step finite difference schemes for the solution of Sturm–Liouville problems

Amodio

Settanni

2015

Communications in Nonlinear Science and Numerical Simulation

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An adaptive optimized Nyström method for second‐order IVPs

Rufai

Mazzia

Ramos

2022

Math Methods in App Sciences

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This research work deals with the development, analysis, and implementation of an adaptive optimized one-step Nyström method for solving second-order initial value problems of ODEs and time-dependent partial differential equations. The new method is developed through a collocation technique with a new approach for selecting the collocation points. An embedding-like procedure is used to estimate the error of the proposed optimized method. The current approach has been used to compute efficiently approximate solutions to general second-order IVPs. The numerical experiments demonstrate that the introduced error estimation and step-size control strategy presented in this manuscript have produced a good performance compared to some of the other existing numerical methods. KEYWORDS collocation method, error estimation and control, ordinary and time-dependent partial differential equations, optimized Nyström method, variable stepsize formulation MSC CLASSIFICATION 65L05, 65L20

show abstract

Asymptotical computations for a model of flow in saturated porous media

Cited by 9 publications

References 15 publications

Multilayer Asymptotic Solution for Wetting Fronts in Porous Media with Exponential Moisture Diffusivity

Multilayer Asymptotic Solution for Wetting Fronts in Porous Media with Exponential Moisture Diffusivity

Reprint of Variable-step finite difference schemes for the solution of Sturm–Liouville problems

An adaptive optimized Nyström method for second‐order IVPs

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