Method of lines

Hamdi, Samir; Schiesser, William E.; Griffiths, Graham W.

doi:10.4249/scholarpedia.2859

Cited by 91 publications

(64 citation statements)

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“…In addition, we have used empirically determined shape functions [4], [7][8][9] for the pressure and shear.…”

Section: Resultsmentioning

confidence: 99%

“…This method is described in detail in [9], which also provides finite-difference formulae that are of fourth-order (or higher) accuracy for the spatial derivatives. Given a set of values for h(x, t) at any time at a discrete set of x values in (19) with (20) and (21), and inputing the known values of P (x), G(x) and S and their derivatives, an expression for h t can be computed at each point using the fourth-order formulae of [9]. These can then be integrated forward in time using Octave's fourth-order lsode routine from an initial state h(x, 0) = h • with some perturbation (if necessary) as described below.…”

Section: Numerical Studiesmentioning

confidence: 99%

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Deformations during jet-stripping in the galvanizing process

et al. 2010

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The problem of coating steel by passing it through a molten alloy and then stripping off excess coating using an air jet is considered. The work revisits the analysis of Tuck (Phys Fluids 26:2352, 1983) with the addition of extra shear terms and a consideration of the effect of increased air-jet speeds. A first-order partial differential equation is derived and solved both to obtain the steady-state coating shape and to determine the evolution of any defects that may form.

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“…In addition, we have used empirically determined shape functions [4], [7][8][9] for the pressure and shear.…”

Section: Resultsmentioning

confidence: 99%

Section: Numerical Studiesmentioning

confidence: 99%

Deformations during jet-stripping in the galvanizing process

et al. 2010

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“…The Method of Lines [Liskovets, 1965;Hamdi et al 2007] (MoL) was used to convert a vertically discretized 1-D Richards' equation to a system of ODEs by Lee et al [2005], and applied by Fatichi et al [2012]. The following sections discuss our use of the MoL in the context of our h-discretization to convert the partial derivatives in equation (11) into finite difference forms, resulting in a set of three ODEs.…”

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confidence: 99%

A new general 1‐D vadose zone flow solution method

Ogden

Lai

Steinke

et al. 2015

Water Resources Research

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We have developed an alternative to the one-dimensional partial differential equation (PDE) attributed to Richards (1931) that describes unsaturated porous media flow in homogeneous soil layers. Our solution is a set of three ordinary differential equations (ODEs) derived from unsaturated flux and mass conservation principles. We used a hodograph transformation, the Method of Lines, and a finite water-content discretization to produce ODEs that accurately simulate infiltration, falling slugs, and groundwater table dynamic effects on vadose zone fluxes. This formulation, which we refer to as ''finite water-content'', simulates sharp fronts and is guaranteed to conserve mass using a finite-volume solution. Our ODE solution method is explicitly integrable, does not require iterations and therefore has no convergence limits and is computationally efficient. The method accepts boundary fluxes including arbitrary precipitation, bare soil evaporation, and evapotranspiration. The method can simulate heterogeneous soils using layers. Results are presented in terms of fluxes and water content profiles. Comparing our method against analytical solutions, laboratory data, and the Hydrus-1D solver, we find that predictive performance of our finite water-content ODE method is comparable to or in some cases exceeds that of the solution of Richards' equation, with or without a shallow water table. The presented ODE method is transformative in that it offers accuracy comparable to the Richards (1931) PDE numerical solution, without the numerical complexity, in a form that is robust, continuous, and suitable for use in large watershed and land-atmosphere simulation models, including regional-scale models of coupled climate and hydrology.

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“…This approach is also called method of lines (see, e.g. [3]). We discretize space into N equal size grid cells of size h = A/N, and define x i = h/2 + i h, so that x i is the center of cell I i =…”

Section: Semi-discrete Approximationmentioning

confidence: 99%