Impact of diffusion coefficient averaging on solution accuracy of the 2D nonlinear diffusive wave equation for floodplain inundation

Gąsiorowski, Dariusz

doi:10.1016/j.jhydrol.2014.06.039

Cited by 9 publications

(9 citation statements)

References 30 publications

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“…where i is the index of the internal node; i = 2,3,...,M−1,M is the total number of nodes; n is the index of the time level; ∆z is the space step; ∆t is the time step; θ n i is the nodal value of the water content; h n i is the nodal value of the pressure head; K n i is the nodal value of hydraulic conductivity; K n i+1/2 is the nodal value of hydraulic conductivity averaged using the arithmetic mean [36][37][38], K n i+1/2 = 0.5(K i + K i+1 ); b is a parameter; b = 1 if the advection term is incorporated into the equation for z direction, otherwise b = 0; and η is the weighting parameter ranging from 0 to 1.…”

Section: Solution Of 1d Equationsmentioning

confidence: 99%

Numerical Solution of the Two-Dimensional Richards Equation Using Alternate Splitting Methods for Dimensional Decomposition

Gąsiorowski

Kolerski

2020

Water

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Research on seepage flow in the vadose zone has largely been driven by engineering and environmental problems affecting many fields of geotechnics, hydrology, and agricultural science. Mathematical modeling of the subsurface flow under unsaturated conditions is an essential part of water resource management and planning. In order to determine such subsurface flow, the two-dimensional (2D) Richards equation can be used. However, the computation process is often hampered by a high spatial resolution and long simulation period as well as the non-linearity of the equation. A new highly efficient and accurate method for solving the 2D Richards equation has been proposed in the paper. The developed algorithm is based on dimensional splitting, the result of which means that 1D equations can be solved more efficiently than as is the case with unsplit 2D algorithms. Moreover, such a splitting approach allows any algorithm to be used for space as well as time approximation, which in turn increases the accuracy of the numerical solution. The robustness and advantages of the proposed algorithms have been proven by two numerical tests representing typical engineering problems and performed for typical properties of soil.

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Section: Solution Of 1d Equationsmentioning

confidence: 99%

Numerical Solution of the Two-Dimensional Richards Equation Using Alternate Splitting Methods for Dimensional Decomposition

Gąsiorowski

Kolerski

2020

Water

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“…For simplicity, in Equation 7and in other algebraic equations, a second subscript for direction y was omitted. The nodal value of the average diffusion coefficient K j can be estimated as follows [40].…”

Section: Solution Of the 1d Equation Using The Fdm Explicit Schemementioning

confidence: 99%

“…The stability analysis performed by the Neumann method and the accuracy analysis carried out using the modified equation approach [42] showed that ω > 1/2 reduces non-physical oscillations related with numerical dispersity [40,44]. The parameter θ determines the accuracy and the stability of time integration, and particular values of this parameter lead to some well-known methods of solving ODE.…”

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confidence: 99%

Computationally Efficient Solution of a 2D Diffusive Wave Equation Used for Flood Inundation Problems

Artichowicz¹,

Gąsiorowski²

2019

Water

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This paper presents a study dealing with increasing the computational efficiency in modeling floodplain inundation using a two-dimensional diffusive wave equation. To this end, the domain decomposition technique was used. The resulting one-dimensional diffusion equations were approximated in space with the modified finite element scheme, whereas time integration was carried out using the implicit two-level scheme. The proposed algorithm of the solution minimizes the numerical errors and is unconditionally stable. Consequently, it is possible to perform computations with a significantly greater time step than in the case of the explicit scheme. An additional efficiency improvement was achieved using the symmetry of the tridiagonal matrix of the arising system of nonlinear equations, due to the application of the parallelization strategy. The computational experiments showed that the proposed parallel implementation of the implicit scheme is very effective, at about two orders of magnitude with regard to computational time, in comparison with the explicit one.

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“…Originally, such an approach has been proposed by Szymkiewicz (1995) for transport and unsteady flow equations in an open channel. The modified FEM has also been developed for the simulation of floodplain inundation by the 2D DWE (Szymkiewicz and Gąsiorowski 2012), as well as for the 1D DWE (Gąsiorowski 2013(Gąsiorowski , 2014. In the modified FEM approach for the 1D case, where the domain of length L with a structure of M − 1 linear elements is assumed ( Fig.…”

Section: Numerical Solution Of 1d Diffusive Equationmentioning

confidence: 99%

“…In the latter situation an inaccurate approximation by the standard arithmetic average leads to a unphysical solution, which manifests itself in an excessive accumulation of water volume in the vicinity of the obstacle. One of possible approaches providing satisfactory results for such hydrodynamic conditions is the arithmetic average based on the previously calculated nodal values of the diffusion coefficient (Gąsiorowski 2014):…”

Section: Numerical Solution Of 1d Diffusive Equationmentioning

confidence: 99%

Modelling of Flood Wave Propagation with Wet-dry Front by One-dimensional Diffusive Wave Equation

Gąsiorowski

2014

Archives of Hydro-Engineering and Environmental Mechanics

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A full dynamic model in the form of the shallow water equations (SWE) is often useful for reproducing the unsteady flow in open channels, as well as over a floodplain. However, most of the numerical algorithms applied to the solution of the SWE fail when flood wave propagation over an initially dry area is simulated. The main problems are related to the very small or negative values of water depths occurring in the vicinity of a moving wet-dry front, which lead to instability in numerical solutions. To overcome these difficulties, a simplified model in the form of a non-linear diffusive wave equation (DWE) can be used. The diffusive wave approach requires numerical algorithms that are much simpler, and consequently, the computational process is more effective than in the case of the SWE. In this paper, the numerical solution of the one-dimensional DWE based on the modified finite element method is verified in terms of accuracy. The resulting solutions of the DWE are compared with the corresponding benchmark solution of the one-dimensional SWE obtained by means of the finite volume methods. The results of numerical experiments show that the algorithm applied is capable of reproducing the reference solution with satisfactory accuracy even for a rapidly varied wave over a dry bottom.

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Impact of diffusion coefficient averaging on solution accuracy of the 2D nonlinear diffusive wave equation for floodplain inundation

Cited by 9 publications

References 30 publications

Numerical Solution of the Two-Dimensional Richards Equation Using Alternate Splitting Methods for Dimensional Decomposition

Numerical Solution of the Two-Dimensional Richards Equation Using Alternate Splitting Methods for Dimensional Decomposition

Computationally Efficient Solution of a 2D Diffusive Wave Equation Used for Flood Inundation Problems

Modelling of Flood Wave Propagation with Wet-dry Front by One-dimensional Diffusive Wave Equation

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