Dimensional Advection Diffusion Equation Research Articles

Numerical Simulation of One-Dimensional Advection Diffusion Equation by New Hybrid Explicit Finite Difference Schemes

In this paper, the aim is to develop two Hybrid Explicit Schemes based on the Finite Difference method for a one- dimensional Advection Diffusion Equation. Moreover, the study considered the advection-diffusion equation as an initial boundary value problem (IBVP) for numerical solutions obtained from various second-order explicit methods along with the solution by proposed methods. Von-Neumann stability analysis is used to analyze the stability of the developed schemes graphically. In the numerical analysis of errors, the L 2 has been computed to compare proposed methods with existing methods in literature and has been carried out in different conditions and step sizes. Proposed methods are robust explicit methods for the purpose of solution of the 1-D advection-diffusion equation.

Mathematical analysis of a Sips-based model for column adsorption

We investigate the dynamics of a contaminated fluid flowing through an adsorption column. We derive a one dimensional advection–diffusion equation coupled to a sink term that accounts for the contaminant adsorption. The adsorption rate is modelled by the Sips equation, where the order of the exponents is obtained through analysing the chemical reaction (as opposed to data fitting). By applying a travelling wave substitution the governing equations are reduced to two coupled ordinary differential equations. After non-dimensionalising and imposing typical values for the operating parameters we are able to identify negligible terms and so reduce the system to one where explicit solutions can be obtained. Distinct solution forms are provided for a range of Sips exponents. The approximate solutions are verified against numerics and experimental data. It is shown that if the breakthrough data is plotted in the form suggested by the analytical solutions then the result is a straight line when the correct Sips exponents are employed. This provides a clear indicator of the correct adsorption model. The straight line form permits a trivial optimisation procedure to determine the adsorption rate. This is the first time that analytical solutions have been obtained for sink terms beyond the standard Langmuir and Linear Driving Force models.

A reduced model for solute transport in compliant blood vessels with arbitrary axial velocity profile

The main result of this paper is the formulation and derivation of a solute transport model in the case of a general velocity profile and in a blood vessel of varying cross section. We couple the resulting one dimensional advection-diffusion equation to a reduced model for blood flow in a compliant vessel, and we obtain a system of equations describing the evolution of the vessel area, blood momentum, and radially averaged solute concentration. In the special case of a no-slip axial velocity profile, we specify a relation between the diffusion coefficient and the shape of the profile. This relation is investigated numerically with physiological boundary data for blood flow, and the results show that the Coriolis parameter has an impact in the transient regime of solute transport.

Two Dimension Vertically Integrated Numerical Suspended Sediment Transport Model (Dept.C)

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Finite Difference Formulation for Prediction of Water Pollution

Water is an important component of the earth. Human being and living organisms are demand for the quality of water. Human activity is one of the causes of the water pollution. The pollution happened give bad effect to the physical and characteristic of water contents. It is not practical to monitor all aspects of water flow and transport distribution. So, in order to help people to access to the polluted area, a prediction of water pollution concentration must be modelled. This study proposed a one-dimensional advection diffusion equation for predicting the water pollution concentration transport. The numerical modelling will be produced in order to predict the transportation of water pollution concentration. In order to approximate the advection diffusion equation, the implicit Crank Nicolson is used. For the purpose of the simulation, the boundary condition and initial condition, the spatial steps and time steps as well as the approximations of the advection diffusion equation have been encoded. The results of one dimensional advection diffusion equation have successfully been used to predict the transportation of water pollution concentration by manipulating the velocity and diffusion parameters.

The effect of uncertain geometries on advection\u2013diffusion of scalar quantities

The two dimensional advection–diffusion equation in a stochastically varying geometry is considered. The varying domain is transformed into a fixed one and the numerical solution is computed using a high-order finite difference formulation on summation-by-parts form with weakly imposed boundary conditions. Statistics of the solution are computed non-intrusively using quadrature rules given by the probability density function of the random variable. As a quality control, we prove that the continuous problem is strongly well-posed, that the semi-discrete problem is strongly stable and verify the accuracy of the scheme. The technique is applied to a heat transfer problem in incompressible flow. Statistical properties such as confidence intervals and variance of the solution in terms of two functionals are computed and discussed. We show that there is a decreasing sensitivity to geometric uncertainty as we gradually lower the frequency and amplitude of the randomness. The results are less sensitive to variations in the correlation length of the geometry.

Computational solutions of three-dimensional advection-diffusion equation using fourth order time efficient alternating direction implicit scheme

To develop an efficient numerical scheme for three-dimensional advection diffusion equation, higher order ADI method was proposed. 2nd and fourth order ADI schemes were used to handle such problem. Von Neumann stability analysis shows that Alternating Direction Implicit scheme is unconditionally stable. The accuracy and efficiency of such schemes was depicted by two test problems. Numerical results for two test problems were carried out to establish the performance of the given method and to compare it with the others Typical methods. Fourth order ADI method were found to be very efficient and stable for solving three dimensional Advection Diffusion Equation. The proposed methods can be implemented for solving non-linear problems arising in engineering and physics.

Numerical Solutions Based on Finite Difference Techniques for Two Dimensional Advection-Diffusion Equation

Numerical Solutions Based on Finite Difference Techniques for Two Dimensional Advection-Diffusion Equation

Chebyshev–Gauss–Lobatto Pseudo–spectral Method for One–Dimensional Advection–Diffusion Equation with Variable coefficients

Chebyshev–Gauss–Lobatto Pseudo–spectral Method for One–Dimensional Advection–Diffusion Equation with Variable coefficients

Numerical simulation of advection-diffusion equation

This paper presents numerical solution of advection-diffusion equation (ADE) using B-spline finite element method. The one dimensional ADE describes many quantities such as vorticity, heat, mass, energy, velocity etc. Test example shows the accuracy of the numerical technique where different comparisons are made for the exact and approximate solution obtained. Numerical results are found to be in good agreement with the analytical results.

Nonlinear Waves in Capillary Electrophoresis

Electrophoretic separation of a mixture of chemical species is a fundamental technique of great usefulness in biology, health care, and forensics. In capillary electrophoresis, the sample migrates in a microcapillary in the presence of a background electrolyte. When the ionic concentration of the sample is sufficiently high, the signal is known to exhibit features reminiscent of nonlinear waves including sharp concentration "shocks." In this paper, we consider a simplified model consisting of a single sample ion and a background electrolyte consisting of a single coion and a counterion in the absence of any processes that might change the ionization states of the constituents. If the ionic diffusivities are assumed to be the same for all constituents the concentration of sample ion is shown to obey a one dimensional advection diffusion equation with a concentration dependent advection velocity. If the analyte concentration is sufficiently low in a suitable nondimensional sense, Burgers' equation is recovered, and thus the time dependent problem is exactly solvable with arbitrary initial conditions. In the case of small diffusivity, either a leading edge or trailing edge shock is formed depending on the electrophoretic mobility of the sample ion relative to the background ions. Analytical formulas are presented for the shape, width, and migration velocity of the sample peak and it is shown that axial dispersion at long times may be characterized by an effective diffusivity that is exactly calculated. These results are consistent with known observations from physical and numerical simulation experiments.

Space–time Galerkin least-squares method for the one–dimensional advection–diffusion equation

The advection–diffusion equation has a long history as a benchmark for numerical methods. The standard Galerkin finite-element method is accurate but lacks sufficient stability, whereas the least-squares method is stable but not accurate for small values of diffusion coefficient. One possibility to overcome it is to use a stabilized finite-element method such as Galerkin least-squares and this is the standard approach taken in this paper. The space–time GLS has been constructed by using both linear and quadratic B-spline shape functions. If advection dominates over diffusion the numerical solution is difficult, especially if boundary layers are to be resolved. The design of stability parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show that the behaviour of the method with emphasis on treatment of boundary conditions. Results shown by the method are found to be in good agreement with the exact solution.

Dispersion in non‐ideal packed beds

AbstractThis work reanalyzes published time series dispersion traces (Han et al., AIChE J. 1985;31:277–288) from step input tracer passages recorded at different locations along the length of a packed bed filled with monodisperse solid spheres. The intrinsic dispersivity and sample dependent dispersion are separated by imposing a heuristic model where coarse grained axial advection velocities vary in a plane perpendicular to the flow but not along the direction of flow. The derived intrinsic dispersivity agrees with the predicted value (Scheven et al., Phys Rev Lett. 2007;99:054502‐1–054502‐4). It serves as a bench mark for different implementations of simulations coupling Stokes flow and diffusion in random geometries, and for experimental tests of injection and packing methods. Conceptually, a well defined and fittable effective dispersivity is introduced in an analytical framework describing dispersion data obtained in non‐ideal packed beds, where elution profiles cannot be fitted to the solution of the one dimensional advection diffusion equation. © 2009 American Institute of Chemical Engineers AIChE J, 2010

A third-order upwind scheme for the advection–diffusion equation using spreadsheets

In this study, numerical solution of advection–diffusion equation with third-order upwind scheme by using spreadsheet simulation (ADE-TUSS) is carried out. This is a user-friendly and a flexible solution algorithm for the numerical solution of the one dimensional advection–diffusion equation (ADE). The ADE-TUSS algorithm is based on the description of ADE by using the third-order upwind scheme (TU) for advection term and second-order central finite representation. For the solution of the governing equations, spreadsheet simulation (SS) technique is used instead of conventional solution techniques. The solution of ADE can be obtained for explicit, implicit and Crank–Nicolson schemes by only changing the values of temporal weighted parameter in the ADE-TUSS algorithm. It is clear in the literature that numerical diffusion causes great deviations in the model results when the first- or second-order upwind schemes are used. In order to decrease the numerical diffusion and to obtain oscillation free results, an artificial diffusion term is usually defined or sizes of the time step and/or grid sizes are set small values. Reduce of the grid sizes and/or time step increases the computational time and generally requires writing a fairly complex code when high order schemes are used. However, numerical solution of ADE by taking into account TU scheme has been carried out by using the iterative spreadsheet solution technique in the proposed solution algorithm. In order to simulate transient solution, a simple macro that carries out time cycle is defined with the help of VBA feature of spreadsheets. One of the most important advantages of ADE-TUSS algorithm is that it does not require the matrix algebra at each time step of the transient solutions. In order to test the ADE-TUSS model, two examples having numerical and analytical solutions are solved. Results showed that use of the high-order schemes in the spreadsheet simulation is very applicable for the numerical solution of ADE. Moreover, numerical diffusion problem is drastically prevented by using the ADE-TUSS model.

Nursing together — Australian and Vietnam: trauma nursing in a developing country

An analytical solution of three dimensional advection–diffusion equation, including longitudinal diffusion, for dispersion of air pollutants from a point source at a height close to the surface is used to simulate the field tracer data collected at Idaho Falls, ID, USA, during low wind, stable conditions. The apparent eddy diffusivities are parameterized in terms of downwind distance for near-source dispersion. The model predictions with two different parameterizations (u∗ and σΘ) of turbulence parameters and two different approaches (hourly and segmented plume or short-term averaging) are compared with the observations and the predictions from constant diffusivities solutions (including the standard Gaussian plume solution). The best results are obtained from segmented plume approach (dividing the test period into smaller intervals) with σΘ-parameterization of turbulence parameters. This method uses actual measurements of turbulent wind fluctuations (in terms of σΘ) and the slow but large variability in the wind direction (leading to multiple peaks in the concentration distribution) is accounted for by the segmented plume approach. Simulations based on hourly averaged data are satisfactory especially in terms of peak values. The results from u∗-parameterization of turbulence parameters are not as good as those from σΘ-parameterization because of usage of indirect and rough estimates of u∗ and some empirical relations which may be questionable in low wind conditions.

An analysis of the time integration algorithms for the finite element solutions of incompressible Navier–Stokes equations based on a stabilised formulation

This work is concerned with the analysis of time integration procedures for the stabilised finite element formulation of unsteady incompressible fluid flows governed by the Navier–Stokes equations. The stabilisation technique is combined with several different implicit time integration procedures including both finite difference and finite element schemes. Particular attention is given to the generalised- α method and the linear discontinuous in time finite element scheme. The time integration schemes are first applied to two model problems, represented by a first order differential equation in time and the one dimensional advection–diffusion equation, and subjected to a detailed mathematical analysis based on the Fourier series expansion. In order to establish the accuracy and efficiency of the time integration schemes for the Navier–Stokes equations, a detailed computational study is performed of two standard numerical examples: unsteady flow around a cylinder and flow across a backward facing step. It is concluded that the semi-discrete generalised- α method provides a viable alternative to the more sophisticated and expensive space–time methods for simulations of unsteady flows of incompressible fluids governed by the Navier–Stokes equations.

Simulation of diffusion experiments under light wind, stable conditions by a variable K-theory model

An analytical solution of three dimensional advection–diffusion equation, including longitudinal diffusion, for dispersion of air pollutants from a point source at a height close to the surface is used to simulate the field tracer data collected at Idaho Falls, ID, USA, during low wind, stable conditions. The apparent eddy diffusivities are parameterized in terms of downwind distance for near-source dispersion. The model predictions with two different parameterizations ( u ∗ and σ Θ) of turbulence parameters and two different approaches (hourly and segmented plume or short-term averaging) are compared with the observations and the predictions from constant diffusivities solutions (including the standard Gaussian plume solution). The best results are obtained from segmented plume approach (dividing the test period into smaller intervals) with σ Θ-parameterization of turbulence parameters. This method uses actual measurements of turbulent wind fluctuations (in terms of σ Θ) and the slow but large variability in the wind direction (leading to multiple peaks in the concentration distribution) is accounted for by the segmented plume approach. Simulations based on hourly averaged data are satisfactory especially in terms of peak values. The results from u ∗ -parameterization of turbulence parameters are not as good as those from σ Θ-parameterization because of usage of indirect and rough estimates of u ∗ and some empirical relations which may be questionable in low wind conditions.

On Destabilizing Implicit Factors in Discrete Advection-Diffusion Equations

In the present paper, we find necessary and sufficient stability conditions for a simple one-time step finite difference discretization of an N -dimensional advection diffusion equation. Furthermore, it is shown that when the implicit factors differ in each direction, a strange behavior occurs: By increasing one implicit factor in only one direction, a stable scheme can become unstable. It is thus suggested to use a single implicit direction (for efficient computing), or the same implicit factor in each direction.