The complete flux scheme—Error analysis and application to plasma simulation

Liu, Lei L; Dijk, van J Jan; Boonkkamp, ten Jhm Jan Thije; Mihailova, DB Diana; Mullen, van der Jjam Joost

doi:10.1016/j.cam.2013.03.011

Cited by 22 publications

(17 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This allows us to use much coarser grid for the CF scheme than would be required for the central difference scheme. Moreover, we can show that the inclusion of the inhomogeneous term guarantees second-order convergence in the limit | | → ∞; see [17]. Next, we consider the time-dependent version of the conservation law, which reads…”

Section: Numerical Scheme For the Groundwater Modelmentioning

confidence: 99%

A Finite Volume-Complete Flux Scheme for a Polluted Groundwater Site

Eikelder

Boonkkamp

Moonen

et al. 2016

Advances in Numerical Analysis

View full text Add to dashboard Cite

We present a model of a polluted groundwater site. The model consists of a coupled system of advection-diffusion-reaction equations for the groundwater level and the concentration of the pollutant. We use the complete flux scheme for the space discretization in combination with the ϑ-method for time integration and we prove a new stability result for the scheme. Numerical results are computed for the Guarani Aquifer in South America and they show good agreement with results in literature.

show abstract

Section: Numerical Scheme For the Groundwater Modelmentioning

confidence: 99%

A Finite Volume-Complete Flux Scheme for a Polluted Groundwater Site

Eikelder

Boonkkamp

Moonen

et al. 2016

Advances in Numerical Analysis

View full text Add to dashboard Cite

show abstract

“…Based on the work reported by Thiart (1990a, 1990b), ten Thije Boonkkamp and Anthonissen (2011) and Liu et al (2013) developed an exponential finite volume-complete flux method in which the flux at the control volume’s boundaries was determined by solving analytically local two-point boundary value problems characterized by piecewise linear coefficients and accounted for the source term. A piecewise linear approximation for the coefficients of one-dimensional (1D) advection-diffusion equations has been previously used by El-Mistikawy and Werle (1978) who developed an exponentially fitted method that was more accurate than those proposed by de G. Allen and Southwell (1955), Il’in (1969) and Scharfetter and Gummel (1969).…”

Section: Introductionmentioning

confidence: 99%

A conservative method of lines for advection-reaction-diffusion equations

Ramos

2020

HFF

View full text Add to dashboard Cite

Purpose The purpose of this study is to develop a new method of lines for one-dimensional (1D) advection-reaction-diffusion (ADR) equations that is conservative and provides piecewise analytical solutions in space, compare it with other finite-difference discretizations and assess the effects of advection and reaction on both 1D and two-dimensional (2D) problems. Design/methodology/approach A conservative method of lines based on the piecewise analytical integration of the two-point boundary value problems that result from the local solution of the advection-diffusion operator subject to the continuity of the dependent variables and their fluxes at the control volume boundaries is presented. The method results in nonlinear first-order, ordinary differential equations in time for the nodal values of the dependent variables at three adjacent grid points and triangular mass and source matrices, reduces to the well-known exponentially fitted techniques for constant coefficients and equally spaced grids and provides continuous solutions in space. Findings The conservative method of lines presented here results in three-point finite difference equations for the nodal values, implicitly treats the advection and diffusion terms and is unconditionally stable if the reaction terms are implicitly treated. The method is shown to be more accurate than other three-point, exponentially fitted methods for nonlinear problems with interior and/or boundary layers and/or source/reaction terms. The effects of linear advection in 1D reacting flow problems indicates that the wave front steepens as it approaches the downstream boundary, whereas its back corresponds to a translation of the initial conditions; for nonlinear advection, the wave front exhibits steepening but the wave back shows a linear dependence on space. For a system of two nonlinearly coupled, 2D ADR equations, it is shown that a counter-clockwise rotating vortical field stretches the spiral whose tip drifts about the center of the domain, whereas a clock-wise rotating one compresses the wave and thickens its arms. Originality/value A new, conservative method of lines that implicitly treats the advection and diffusion terms and provides piecewise-exponential solutions in space is presented and applied to some 1D and 2D advection reactions.

show abstract

“…From the current literature on CFS, one can easily find that apart from the recent contribution of Ten Thije Boonkkamp and a few of his co‐authors , there are hardly any reportings related to the work based on CFS. Hence, here an attempt has been made to explore the effectiveness of CFS in capturing the boundary layers associated with the parabolic SPDDEs for the first time.…”

Section: Introductionmentioning

confidence: 99%

Complete flux scheme for parabolic singularly perturbed differential‐difference equations

Kumar

Boonkkamp

2018

Numerical Methods Partial

View full text Add to dashboard Cite

In this study, we investigate the concept of the complete flux (CF) obtained as a solution to a local boundary value problem (BVP) for a given parabolic singularly perturbed differential‐difference equation (SPDDE) with modified source term to propose an efficient complete flux‐finite volume method (CF‐FVM) for parabolic SPDDE which is μ‐ and ϵ‐uniform method where μ, ϵ are shift and perturbation parameters, respectively. The proposed numerical method is shown to be consistent, stable, and convergent and has been successfully implemented on three test problems.

show abstract

The complete flux scheme—Error analysis and application to plasma simulation

Cited by 22 publications

References 18 publications

A Finite Volume-Complete Flux Scheme for a Polluted Groundwater Site

A Finite Volume-Complete Flux Scheme for a Polluted Groundwater Site

A conservative method of lines for advection-reaction-diffusion equations

Complete flux scheme for parabolic singularly perturbed differential‐difference equations

Contact Info

Product

Resources

About