An adaptive local grid refinement based on the exact peak capture and oscillation free scheme to solve transport equations

Yeh, Gour Tsyh; Chang, Jing Ru; Cheng, Hsu-Tang; Sung, Chao Ho

doi:10.1016/0045-7930(94)00033-u

Cited by 20 publications

(12 citation statements)

References 44 publications

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“…However, the ability of these methods to effectively deal with peak attenuation has not been addressed. For example, conventional Eulerian based methods, such as QUICKEST (Leonard 1979), ULTIMATE (Leonard and Mokhtari 1990)-one of the best TVD methods, and ENO (Shu and Osher 1988;Rao 2005), generate about 30% peak clipping when used to solve a one-dimensional benchmark problem (Yeh et al 1995). Although, some modifications of these methods have been reported, such as WENO (Weighted ENO) (Jiang and Shu 1996;Aràndiga and Velda 2004;Bryson and Levy 2006) and WCompact (Weighted Compact) (Deng and Zhang 2000;Jiang et al 2001), which reduced simulation time and oscillations, effects on peak clipping have not been reported.…”

Section: Introductionmentioning

confidence: 98%

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An adaptive local grid refinement and peak/valley capture algorithm to solve nonlinear transport problems with moving sharp-fronts

et al. 2007

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Highly nonlinear advection-dispersion-reaction equations govern numerous transport phenomena. Robust, accurate, and efficient algorithms to solve these equations hold the key to the success of applying numerical models to field problems. This paper presents the development and verification of a computational algorithm to approximate the highly nonlinear transport equations of reactive chemical transport and multiphase flow. The algorithm was developed based on the Lagrangian-Eulerian decoupling method with an adaptive ZOOMing and Peak/valley Capture (LEZOOMPC) scheme. It consists of both backward and forward node tracking, rough element determination, peak/valley capturing, and adaptive local grid refinement. A second-order tracking was implemented to accurately and efficiently track all fictitious particles. Shanks' method was introduced to deal with slowly converging case. The accuracy and efficiency of the algorithm were verified with the Burger equation for a variety of cases. The robustness of the algorithm to achieve convergent solutions was demonstrated by highly nonlinear reactive contaminant transport and multiphase flow problems.

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Section: Introductionmentioning

confidence: 98%

“…(2) forward node tracking to determine rough elements and to capture peaks/valleys; and (3) local refinement made at forward-tracked nodes in rough elements (Yeh et al 1995).…”

Section: Introductionmentioning

confidence: 99%

An adaptive local grid refinement and peak/valley capture algorithm to solve nonlinear transport problems with moving sharp-fronts

et al. 2007

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“…Equation (24) shows that now also corrections are implemented for the Darcian fluid flux density, q. This result is different from, and more accurate than, the approach originally used by van Genuchten for solute transport under transient variably saturated water flow condition.…”

Section: Third-order Approximation In Timementioning

confidence: 83%

A Third-Order Numerical Scheme With Upwind Weighting for Solving the Solute Transport Equation

Huang

Šimůnek

Genuchten

1997

Int. J. Numer. Meth. Engng.

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SUMMARYSolute transport in the subsurface is generally described quantitatively with the convection-dispersion transport equation. Accurate numerical solutions of this equation are important to ensure physically realistic predictions of contaminant transport in a variety of applications. An accurate third-order in time numerical approximation of the solute transport equation was derived. The approach leads to corrections for both the dispersion coefficient and the convective velocity when used in numerical solutions of the transport equation. The developed algorithm is an extension of previous work to solute transport conditions involving transient variably saturated fluid flow and non-linear adsorption. The third-order algorithm is shown to yield very accurately solutions near sharp concentration fronts, thereby showing its ability to eliminate numerical dispersion. However, the scheme does suffer from numerical oscillations. The oscillations could be avoided by employing upwind weighting techniques in the numerical scheme. Solutions obtained with the proposed method were free of numerical oscillations and exhibited negligible numerical dispersion. Results for several examples, including those involving highly non-linear sorption and infiltration into initially dry soils, were found to be very accurate when compared to other solutions.1997 by John Wiley & Sons, Ltd.

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“…where C is concentration; N G is the interpolation function at the ith global node; C H is the concentration at node j; DC H /Dt is the total or material derivative of C H with respect to time t; D is the diffusion coefficient tensor; R is the domain of interest; B is the domain boundary; V is the velocity vector; n is the unit vector normal to the domain boundary. It should be noted the velocity term is implicitly included, through the computation of Lagrangian concentrations, in the first term of equation (2). With the finite difference approximation acting on the time derivative term, we can write equation (2) in matrix form as…”

Section: The Computations Of Lezoompcmentioning

confidence: 99%

A Lagrangian-Eulerian method with adaptively local ZOOMing approach to solve three-dimensional advection-diffusion transport equations

Cheng

Yeh

1998

Int. J. Numer. Meth. Engng.

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We present a Lagrangian-Eulerian method with adaptively local ZOOMing and Peak/valley Capturing approach (LEZOOMPC), consisting of advection-diffusion decoupling, backward particle tracking, forward particle tracking, adaptively local zooming, peak/valley capturing, and slave point utilization, to solve three-dimensional advection-diffusion transport equations. This approach and the associated computer code, 3DLEZOOMPC, were developed to circumvent the difficulties associated with the Exact Peak Capturing and Oscillation-Free (EPCOF) scheme, developed earlier by the authors, when it was extended from a one-dimensional space to a three-dimensional space. The accurate results of applying EPCOF to solving two one-dimensional benchmark problems under a variety of conditions have shown the capability of this scheme to eliminate all types of numerical errors associated with the advection term and to keep the maximum computational error to be within the prescribed error tolerance. However, difficulties arose when the EPCOF scheme was extended to a multi-dimensional space mainly due to the geometry. To avoid these geometric difficulties, we modified the EPCOF scheme and named the modified scheme LEZOOMPC. LEZOOMPC uses regularly local zooming for rough elements and peak/valley capturing within subelements to resolve the problems of tetrangulation and boundary source as well as to preserve the shape of concentration distribution. In addition, LEZOOMPC employs the concept of 'slave points' to deal with the compatibility problem in the diffusion zooming of the Eulerian step. As a result, not only is the geometrical problem resolved, but also the spirit of EPCOF is retained. Application of 3DLEZOOMPC to solving an advection-decay and a boundary source benchmark problems indicates its capability in solving advection transport problems accurately to within any prescribed error tolerance by using mesh Courant number ranging from 0 to infinity. Demonstration of using 3DLEZOOMPC to solve an advection-diffusion benchmark problem shows how the numerical solution is improved with the increment of the diffusion zooming factors. 3DLEZOOMPC could solve advection-diffusion transport problems accurately by using mesh Peclet numbers ranging from 0 to infinity and very large time-step size. The size of time-step is related to both the diffusion coefficients and mesh sizes. Hence, it is limited only by the diffusion solver. The application of this approach to a two-dimensional space has been demonstrated earlier in the paper entitled 'A Lagrangian-Eulerian method with adaptively local zooming and peak/valley capturing approach to solve two-dimensional advection-diffusion transport equations'.

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An adaptive local grid refinement based on the exact peak capture and oscillation free scheme to solve transport equations

Cited by 20 publications

References 44 publications

An adaptive local grid refinement and peak/valley capture algorithm to solve nonlinear transport problems with moving sharp-fronts

An adaptive local grid refinement and peak/valley capture algorithm to solve nonlinear transport problems with moving sharp-fronts

A Third-Order Numerical Scheme With Upwind Weighting for Solving the Solute Transport Equation

A Lagrangian-Eulerian method with adaptively local ZOOMing approach to solve three-dimensional advection-diffusion transport equations

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