Finite element methods for time-dependent convection–diffusion–reaction equations with small diffusion

John, Volker; Schmeyer, Ellen

doi:10.1016/j.cma.2008.08.016

Cited by 213 publications

(164 citation statements)

References 47 publications

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“…This smearing caused the growth of unphysically large particles and finally a blow-up of the simulation. Inspired by the bad experiences with the SUPG method and the upwind method, comprehensive numerical studies of stabilized finite element methods for scalar time-dependent convection-diffusion-reaction equations were performed in [17,18]. These studies included besides the SUPG method a number of Spurious Oscillations at Layers Diminishing (SOLD) schemes [11,12], a local projection stabilization (LPS) scheme [25] and two FiniteElement-Method Flux-Corrected-Transport (FEM-FCT) schemes [21,22,23].…”

Section: The Equations For the Chemical Reactionmentioning

confidence: 99%

“…These studies included besides the SUPG method a number of Spurious Oscillations at Layers Diminishing (SOLD) schemes [11,12], a local projection stabilization (LPS) scheme [25] and two FiniteElement-Method Flux-Corrected-Transport (FEM-FCT) schemes [21,22,23]. All studies in [17,18] led to the consistent conclusion that the FEM-FCT schemes are far better than the other approaches. In particular, a linear FEM-FCT scheme from [21] showed a very good relation of accuracy and efficiency.…”

Section: The Equations For the Chemical Reactionmentioning

confidence: 99%

“…equations of the form (16) with ε = r = f = 0. An extension to convectiondiffusion-reaction equations can be found in [18]. These schemes work on the algebraic level and they modify the system matrix and the right-hand side vector.…”

Section: The Equations For the Chemical Reactionmentioning

confidence: 99%

“…We refer to [18,22] for presentations of this algorithm. Some details on the implementation of FEM-FCT schemes can be found also in [18].…”

Section: The Equations For the Chemical Reactionmentioning

confidence: 99%

“…The simulation of such equations has been studied meanwhile separately in [17,18]. In these studies, much better methods than the standard schemes used in [16] could be identified, see Section 3.2 for details.…”

Section: Introductionmentioning

confidence: 99%

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On the impact of the scheme for solving the higher dimensional equation in coupled population balance systems

John

Roland

2010

Numerical Meth Engineering

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Population balance systems are models for processes in nature and industry which lead to a coupled system of equations (Navier-Stokes equations, transport equations, . . .) where the equations are defined in domains with different dimensions. This paper will study the impact of using different schemes for solving the three-dimensional equation of a precipitation process in a two-dimensional flow domain. The numerical schemes for the three-dimensional equation are assessed with respect to the median of the volume fraction of the particle size distribution and the computational costs. It turns out that in the case of a structured flow field with small variations in time all schemes give qualitatively the same results. For a highly time-dependent flow field, the evolution of the median of the volume fraction differs considerably between first order and higher order schemes.

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Section: The Equations For the Chemical Reactionmentioning

confidence: 99%

Section: The Equations For the Chemical Reactionmentioning

confidence: 99%

Section: The Equations For the Chemical Reactionmentioning

confidence: 99%

“…We refer to [18,22] for presentations of this algorithm. Some details on the implementation of FEM-FCT schemes can be found also in [18].…”

Section: The Equations For the Chemical Reactionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the impact of the scheme for solving the higher dimensional equation in coupled population balance systems

John

Roland

2010

Numerical Meth Engineering

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Numerical stability analysis of two‐dimensional solute transport along a discrete fracture in a porous rock matrix

Watanabe

Kolditz

2015

Water Resources Research

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This work reports numerical stability conditions in two-dimensional solute transport simulations including discrete fractures surrounded by an impermeable rock matrix. We use an advectivedispersive problem described in Tang et al. (1981) and examine the stability of the Crank-Nicolson Galerkin finite element method (CN-GFEM). The stability conditions are analyzed in terms of the spatial discretization length perpendicular to the fracture, the flow velocity, the diffusion coefficient, the matrix porosity, the fracture aperture, and the fracture longitudinal dispersivity. In addition, we verify applicability of the recently developed finite element method-flux corrected transport (FEM-FCT) method by Kuzmin (2009) to suppress oscillations in the hybrid system, with a comparison to the commonly utilized Streamline Upwinding/Petrov-Galerkin (SUPG) method. Major findings of this study are (1) the mesh von Neumann number (Fo) ! 0.373 must be satisfied to avoid undershooting in the matrix, (2) in addition to an upper bound, the Courant number also has a lower bound in the fracture in cases of low dispersivity, and (3) the FEM-FCT method can effectively suppress the oscillations in both the fracture and the matrix. The results imply that, in cases of low dispersivity, prerefinement of a numerical mesh is not sufficient to avoid the instability in the hybrid system if a problem involves evolutionary flow fields and dynamic material parameters. Applying the FEM-FCT method to such problems is recommended if negative concentrations cannot be tolerated and computing time is not a strong issue.

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Population balance modeling of polyurethane foam formation with pressure‐dependent growth kernel

et al. 2021

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Polyurethane foams are widely used materials often chosen for their useful characteristics such as low thermal conductivity, ease of application, and high strength-toweight ratios. Computational models are needed to predict the dynamics of the flow and expansion, and the resulting material properties, to improve manufacturing processes. In this paper, a model for PMDI, a water-blown polyurethane foam, is presented. By extending a kinetics-based approach by adding bubble-scale information via a population balance equation (PBE) using the quadrature method of moments, we can track bubble size distributions during foaming. We present results from a three-dimensional computational fluid dynamics model using arbitrary Lagrangian-Eulerian interface tracking implemented in finite element software. The model compares favorably with experimental data, including dynamics, bubble distributions measured by both camera and diffusion wave spectroscopy, and post-test bubble size from scanning electron microscopy and density measurements from x-ray computed tomography.

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Finite element methods for time-dependent convection–diffusion–reaction equations with small diffusion

Cited by 213 publications

References 47 publications

On the impact of the scheme for solving the higher dimensional equation in coupled population balance systems

On the impact of the scheme for solving the higher dimensional equation in coupled population balance systems

Numerical stability analysis of two‐dimensional solute transport along a discrete fracture in a porous rock matrix

Population balance modeling of polyurethane foam formation with pressure‐dependent growth kernel

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