An Investigation of the Accuracy of the Control-Volume Finite-Element Method Based on Triangular Prismatic Elements for Simulating Diffusion in Anisotropic Media

Truscott, Simon; Turner, Ian

doi:10.1080/10407790490475300

Cited by 14 publications

(14 citation statements)

References 20 publications

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“…In the proposed time-stepping algorithm, the original initial/boundary value problem reduces at each time step to the solution of a boundary value problem defined by the non-homogeneous PDE (12), with the non-homogeneous term given in terms of the solution at the previous time step.…”

Section: Cv-rbf Formulationmentioning

confidence: 99%

“…This technique has been used in the computation of flux corrective terms, Jayantha and Turner [9,10], to increase the spatial accuracy of CV schemes, and also, in a more direct approach, in the reconstruction of the fluxes at the cell faces of the CV [11]. Other researchers proposed the Gauss-Green gradient reconstruction technique (GGRT), which has been used in combination with the LSRT to compute the gradients at the cell faces of the CV [12,13]. New ideas have been found also in the spectral volume (SV) developed by Liu et al [14,15] where the unstructured grid cells are partitioned into structured sub-cells.…”

Section: Introductionmentioning

confidence: 99%

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An implicit upwinding volume element method based on meshless radial basis function techniques for modelling transport phenomena

Orsini

Power

Morvan

et al. 2009

Numerical Meth Engineering

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SUMMARYThis work presents a control volume (CV) method for transient transport problems where the cell surface fluxes are reconstructed using local interpolation functions that besides interpolating the nodal values of the field variable, also satisfies the governing equation at some auxiliary points in the interpolation stencils. The interpolation function relies on a Hermitian radial basis function (HRBF) meshless collocation approach to find the solution of auxiliary local boundary/initial value problems, that are solved using the same time-integration scheme adopted to update the global CV solution. By the use of interpolation functions that approximate the governing equation a form of analytical upwinding scheme is achieved, without the need of using predefined interpolation stencils according to the magnitude and direction of the local advective velocity. In this way, the interpolation formula retains the desired information about the advective velocity field allowing the use of centrally defined stencils even in the cases of advective dominant problems. This new CV approach is referred to as the CV-HRBF method. Two time-stepping formulations are considered: a full implicit approach and the weighted Crank-Nicholson one. The implicit upwinding scheme, intrinsic to the proposed CV-HRBF, is tested by solving a travelling front problem at the Péclet number equal to 500, 1000 and infinity; with the latter corresponding to a shock front. Finally, the accuracy of the numerical method is validated against one-and three-dimensional reactive transport problems characterized by smooth solutions.

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Section: Cv-rbf Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

An implicit upwinding volume element method based on meshless radial basis function techniques for modelling transport phenomena

Orsini

Power

Morvan

et al. 2009

Numerical Meth Engineering

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show abstract

“…This technique has been used in the computation of flux corrective terms (see Turner 2003, 2005) to increase the spatial accuracy of CV schemes, and also, in a more direct approach, in the reconstruction of the fluxes at the cell faces of the control volume (see Ollivier-Gooch and Van Altena 2002). Other researchers proposed the Gauss-Green gradient reconstruction technique (GGRT), which has been used in combination with the LSRT to compute the gradients at the cell faces of the CV (see Truscott and Turner 2004;Manzini and Putti 2007).…”

Section: Introductionmentioning

confidence: 99%

A Control Volume Radial Basis Function Techniques for the Numerical Simulation of Saturated Flows in Semi-confined Aquifer

et al. 2008

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In this work, a Control Volume Radial Basis Function technique (CV-RBF) is adapted to solve ground water flow in the saturated zone of the semi-confined aquifer. The CV-RBF method differs from classical CV methods in the way that the flux at the cell surfaces is computed. A local RBF interpolation of the field variable is performed at the centres of the cell being integrated and its neighbours. This interpolation is then used to reconstruct the solution and its gradient in the integration points which support the flux computation. In addition, it is required that such interpolation satisfies the governing equation in a certain number of points placed around the cell centres. In this way, the local interpolations become equivalent to local boundary-value problems. The CV-RBF method is combined with a local remeshing technique in order to track the phreatic surface, where the gradients required to satisfy the kinematic condition are computed by the same local RBF interpolations used for the flux computation. The proposed numerical approach is validated in a series of three-dimensional groundwater flow problems where the operations of recharging and extracting water from a semi-confined aquifer are modelled.

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“…In the applications which motivated the work reported here, the need arose in approximating the diffusive flux in finite volume discretisation methods for solving conservation equations numerically [4,14] and in the representation of the surfaces of leaves [1,2,10,13] where a smooth fit to scattered data may be required.…”

Section: Introductionmentioning

confidence: 99%

On derivative estimation and the solution of least squares problems

Belward

Turner

Ilić

2008

Journal of Computational and Applied Mathematics

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Surface interpolation finds application in many aspects of science and technology. Two specific areas of interest are surface reconstruction techniques for plant architecture and approximating cell face fluxes in the finite volume discretisation strategy for solving partial differential equations numerically. An important requirement of both applications is accurate local gradient estimation. In surface reconstruction this gradient information is used to increase the accuracy of the local interpolant, while in the finite volume framework accurate gradient information is essential to ensure second order spatial accuracy of the discretisation.In this work two different least squares strategies for approximating these local gradients are investigated and the errors associated with each analysed. It is shown that although the two strategies appear different, they produce the same least squares error. Some carefully chosen case studies are used to elucidate this finding.

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An Investigation of the Accuracy of the Control-Volume Finite-Element Method Based on Triangular Prismatic Elements for Simulating Diffusion in Anisotropic Media

Cited by 14 publications

References 20 publications

An implicit upwinding volume element method based on meshless radial basis function techniques for modelling transport phenomena

An implicit upwinding volume element method based on meshless radial basis function techniques for modelling transport phenomena

A Control Volume Radial Basis Function Techniques for the Numerical Simulation of Saturated Flows in Semi-confined Aquifer

On derivative estimation and the solution of least squares problems

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