Finite difference methods for solving the two‐dimensional advection–diffusion equation

Noye, B. J.; Tan, Hui

doi:10.1002/fld.1650090107

Cited by 103 publications

(108 citation statements)

References 17 publications

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“…The term ''consistent velocity discretization'' is perhaps unfortunate, as the standard discretization method (8) is consistent with usual finite element notions; what Voss [1984] wishes to convey is that equation (9) [Warming and Hyett, 1974;Noye and Hayman, 1985]. However, the MEPDE technique was developed for finite difference methods and has not, to our knowledge, been applied before to a finite element code, such as SUTRA.…”

Section: Sutra's Solution Methodsmentioning

confidence: 99%

“…This often leads to physically unreasonable results and problems with convergence. Another kind of numerical error is numerical dispersion [Noye and Hayman, 1985;Gresho and Sani, 1998]. Numerical dispersion is insidious because it mimics hydrodynamic dispersion (a heuristic description of various physical processes [Bear, 1972]) producing smooth results that may seem plausible.…”

Section: Introductionmentioning

confidence: 99%

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Numerical error in groundwater flow and solute transport simulation

Woods

Teubner

Simmons

et al. 2003

Water Resources Research

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[1] Models of groundwater flow and solute transport may be affected by numerical error, leading to quantitative and qualitative changes in behavior. In this paper we compare and combine three methods of assessing the extent of numerical error: grid refinement, mathematical analysis, and benchmark test problems. In particular, we assess the popular solute transport code SUTRA [Voss, 1984] as being a typical finite element code. Our numerical analysis suggests that SUTRA incorporates a numerical dispersion error and that its mass-lumped numerical scheme increases the numerical error. This is confirmed using a Gaussian test problem. A modified SUTRA code, in which the numerical dispersion is calculated and subtracted, produces better results. The much more challenging Elder problem [Elder, 1967;Voss and Souza, 1987] is then considered. Calculation of its numerical dispersion coefficients and numerical stability show that the Elder problem is prone to error. We confirm that Elder problem results are extremely sensitive to the simulation method used.

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Section: Sutra's Solution Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Numerical error in groundwater flow and solute transport simulation

Woods

Teubner

Simmons

et al. 2003

Water Resources Research

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“…where Ω is a two-dimensional rectangular domain; Γ is the boundary of Ω; analytic solution [41] u(x, y, t) = 1 4t…”

Section: Convection-diffusion Equationsmentioning

confidence: 99%

A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations

Tien

Mai‐Duy

Tran

et al. 2016

Computers & Mathematics with Applications

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In this paper, we propose a simple but effective preconditioning technique to improve the numerical stability of Integrated Radial Basis Function (IRBF) methods. The proposed preconditioner is simply the inverse of a well-conditioned matrix that is constructed using non-flat IRBFs. Much larger values of the free shape parameter of IRBFs can thus be employed and better accuracy for smooth solution problems can be achieved. Furthermore, to improve the accuracy of local IRBF methods, we propose a new stencil, namely Combined Compact IRBF (CCIRBF), in which (i) the starting point is the fourth-order derivative; and (ii) nodal values of first-and second-order derivatives at side nodes of the stencil are included in the computation of first-and second-order derivatives at the middle node in a natural way. The proposed stencil can be employed in uniform/nonuniform Cartesian grids. The preconditioning technique in combination with the CCIRBF scheme employed with large values of the shape parameter are tested with elliptic equations and then applied to simulate several fluid flow problems governed by Poisson, Burgers, convectiondiffusion, and Navier-Stokes equations. Highly accurate and stable solutions are obtained. In some cases, the preconditioned schemes are shown to be several orders of magnitude more accurate than those without preconditioning.

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“…The final implementation in the TASS model however requires the computation of the advection-diffusion equation for various microphysical scalars. For the sake of completeness, the results of the test case for the advection-diffusion equation proposed by Noye and Tan (1989) …”

Section: E Noye-tan Testmentioning

confidence: 99%