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1

An accuracy analysis of the cascaded lattice Boltzmann method for the 1D advection-diffusion equation

Straka Robert, Sharma Keerti V.

Computer Methods in Materials Science

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2020

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Vol. 20, No. 4

173--184

EN

We analyze higher order error terms in a modified partial differential equation of a cascaded lattice Boltzmann method (CLBM) for one conservation law – the advection-diffusion equation. To inspect the behavior of the error terms we derived an equivalent finite difference equation (EFDE), this approach is different from other techniques like the Chapman-Engskog expansion, equivalent partial differential equations or the Maxwell iteration used in the literature. The resulting EFDE is obtained from the recurrence formulas of the lattice Boltzmann equations for the CLBM and is subsequently analyzed by standard analytical techniques. We have found relations of the LBM parameters which could cancel some of the higher order terms, making the method more accurate. The detailed derivation of the EFDE and higher order terms’ pre-factors is the main result of this paper. The resulting explicit form of the error terms are derived and presented.

2

The Dirichlet problem for the time-fractional advection-diffusion equation in a half-space

Povstenko Y., Klekot J.

Journal of Applied Mathematics and Computational Mechanics

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2015

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Vol. 14, nr 2

73--83

EN

The one-dimensional time-fractional advection-diffusion equation with the Caputo time derivative is considered in a half-space. The fundamental solution to the Dirichlet problem and the solution of the problem with constant boundary condition are obtained using the integral transform technique. The numerical results are illustrated graphically.

3

Scenarios of the spread of a waste heat discharge in a river — Vistula River case study

Kalinowska M.B., Rowiński P.M., Kubrak J., Mirosław-Świątek D.

Acta Geophysica

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2012

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Vol. 60, no. 1

214-231

EN

The problem of two-dimensional mathematical modelling of heated cooling water discharges into running waters is considered in the paper. Two models — one for the evaluation of 2D turbulent velocity field and the other, developed by authors of the study, for 2D heat transport in open-channels — were used in the calculations. Relevant scenarios of the spread of heated water discharged from a designed gas-stem power plant to be constructed at the Vistula River were presented. Environmentally most friendly variant of the discharge of the thermal pollution was selected from among four various variants.

4

Two-dimensional Model of Fluid Flow and Distribution of Nutrient Concentration in a Microfluidic Device

Wroniszewski P.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2011

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Vol. 15, No 2

135-155

EN

During the past few years, the process of miniaturisation in the field of biochemical laboratory equipment led to the introduction of the so-called "lab-on-a-chip" microdevices, which combine separate functional units into a complex, multifunctional apparatus. Fluid dynamics plays an essential role in the development of such equipment, since frequently the major part of chemical analysis is based on soluble analytes. In this work, we consider a device for the analysis of cell growth under different conditions. In this device, dozens of cell spots absorb the nutrient (analyte) from the liquid medium. The concentration of the analyte must be strictly controlled to maintain a specific microenvironment. A two-dimensional model for the flow field and the distribution of concentration of the analyte is developed taking into account the geometrical shape of the spot with a simplified absorption model. The dependence of the results on the controlling parameters is investigated in order to determine the influence of the presence of the cell spot on the distribution of the analyte.

5

The pollutant transport equation for a steady, gradually varied flow in an open channel network: a solution of high accuracy

Szymkiewicz R.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2007

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Vol. 11, No 4

365-382

EN

The paper is concerned with solving the transport pollutant problem for a steady, gradually varied flow in an open channel network. The 1D advective-diffusive transport equation is solved using the splitting technique. An analytical solution of the linear advective-diffusive equation in the form of an impulse response function is used to solve the advection-diffusion part of the governing equation. This approach, previously applied in solutions of the advection-diffusion equation for a single channel, is extended to a channel network. Numerical calculations are only required to compute the integral of convolution. The finite difference method is used to solve the second part of the governing equation, containing the source term. The applied approach has considerable advantages, especially appreciable in the case of advection-dominated transport with large gradients of concentration, since it generates no numerical dissipation or dispersion. The flow parameters are obtained via solution of the steady, gradually varied flow equation. In the final non-linear system of algebraic equations obtained through approximation of the ordinary differential equation, the depths at each cross-section of channels and the discharge at each branch of the network are considered as unknowns. The system is solved using the modified Picard iteration, which ensures convergence of the iterative process for a steady, gradually varied flow solved for both looped and tree-type open channel networks.

6

Importance of advective zone in longitudinal mixing experiments

Shucksmith J., Boxall J., Guymer I.

Acta Geophysica

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2007

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Vol. 55, no. 1

95-103

EN

One-dimensional Fickian dispersion models such as the advection diffusion equation (ADE) are commonly used to analyse and predict concentration distributions downstream of contamination events in watercourses. Such models are only valid once the tracer had entered the equilibrium zone. This paper compares previous theoretical, experimental and numerical estimates of the distance to reach the equilibrium zone with new experimental values, obtained by examining the change of skewness in a tracer profile, downstream of a cross-sectionally well mixed source. Closer agreement was found with Fischers’ theoretical estimate than prior experimental and numerical studies.