Two-dimensional Advection-diffusion Research Articles

Study of steady two-dimensional advection–diffusion equation with stratification using second-kind shifted Chebyshev polynomials

Study of steady two-dimensional advection–diffusion equation with stratification using second-kind shifted Chebyshev polynomials

Near-field Atmospheric Dispersion Modeling: A New Approach for the Two-dimensional Steady-state Advection-Diffusion Equation Using Fractal Derivative

Near-field Atmospheric Dispersion Modeling: A New Approach for the Two-dimensional Steady-state Advection-Diffusion Equation Using Fractal Derivative

Towards intermediate complexity modelling of contrail formation: the new dynamical framework RadMod

Abstract Contrails are a major contributor to the climate effect of aviation. Mitigation efforts and technological improvements aim to reduce the contrail climate effect. Many currently discussed innovations (like using sustainable aviation fuels (SAFs) or hydrogen) affect the physical processes and phenomena during contrail formation. Hence, understanding and analysing contrail formation is of great importance in the context of climate research. Ice crystal formation in a nascent contrail is completed within the first seconds after the engine exhaust is emitted. In the past, numerical models treating this early stage typically involved either a 3D or 0D approach. Whereas 3D models are computationally expensive, restricting the number of simulations that could be performed, less expensive 0D models allow to explore a larger parameter space but neglect plume heterogeneity and use a prescribed plume dilution. We present the new dynamical framework RadMod for contrail formation simulations that describes the evolution of a turbulent round jet emitted from an aircraft engine. Relative to large-eddy simulation (LES) or Reynolds-averaged Navier-Stokes (RANS) 3D models of contrail formation, our model is computationally less expensive, enabling extensive parameter studies. The model accounts for the mixing of the hot and moist exhaust air with the cold ambient air through the solution of the two-dimensional advection-diffusion equation of momentum, temperature, and water vapour. The validation of our model is conducted through comparisons with empirical relationships and CFD results. In the near future, this model will be combined with an existing microphysical model, resulting in a contrail formation model of intermediate complexity.

Two-Dimensional Time Fractional River-Pollution Model and Its Remediation by Unsteady Aeration

This study contains a mathematical model for river pollution and its remediation for an unsteady state and investigates the effect of aeration on the degradation of pollutants. The governing equation is a pair of nonlinear time-fractional two-dimensional advection-diffusion equations for pollutant and dissolved oxygen (DO) concentration. The coupling of these equations arises due to the chemical interactions between oxygen and pollutants, forming harmless chemicals. The Fractional Reduced Differential Transform Method (FRDTM) is applied to provide approximate solutions for the given model. Also, the convergence of solutions is checked for efficacy and accuracy. The effect of longitudinal and transverse diffusion coefficients of pollutant and DO on the concentration of pollutant and DO is analyzed numerically and graphically. Also, we checked the effect of change in the river’s longitudinal and transverse seepage velocity on pollutant and DO concentration numerically and graphically. We analyzed the comparison of change in the value of half-saturated oxygen demand concentration for pollutant decay on pollutant and DO concentration numerically and graphically. Also, numerical and graphical analysis examined the effect of fractional parameters on pollution levels.

A sixth-order finite difference method for the two-dimensional nonlinear advection diffusion reaction equation

A novel higher-order finite difference method is developed to solve the two-dimensional nonlinear advection diffusion reaction equation. Firstly, a new five-point sixth-order finite difference scheme is proposed to discretize spatial derivatives and the sixth-order backward difference method is applied to teat temporal derivative. As a result, a new fully implicit scheme with sixth-order accuracy in both time and space is proposed. Subsequently, for the calculation of the starting step of the sixth-order backward difference method, the Crank-Nicolson method combined with the Richardson extrapolation technique is employed to obtain the equivalent temporal accuracy as the main scheme. Finally, the accuracy and reliability of the presented method are validated through various numerical examples.

On the sediment deposition on Barugbug Situdam Weir

In this paper, we investigate the behavior of total sediment transport in the Barudbug weir, Situdam, Karawang, West Java, Indonesia, which is a crucial aspect of water management in the Citarum river basin. We utilize SESAME instrument observation data, encompassing rainfall, TSS, and depth parameters, gathered over a period of approximately three years. Additionally, we employ a two-dimensional Advection-Diffusion analytic model that takes into account settling velocity. The observation results reveal a positive correlation between TSS and depth, with TSS levels increasing following heavy rainfall. Furthermore, there exists a time delay between rainfall events and the subsequent rise in TSS concentration, suggesting an upstream source of sediment. The model outcomes demonstrate an inverse relationship between TSS concentration and the velocity of river flow, as the river flow acts as a leaching mechanism.

Numerical Solution of Two-Dimensional Nonlinear Unsteady Advection-Diffusion-Reaction Equations with Variable Coefficients

The advection-diffusion-reaction (ADR) equation is a fundamental mathematical model used to describe various processes in many different areas of science and engineering. Due to wide applicability of the ADR equation, finding accurate solution is very important to better understand a physical phenomenon represented by the equation. In this study, a numerical scheme for solving two-dimensional unsteady ADR equations with spatially varying velocity and diffusion coefficients is presented. The equations include nonlinear reaction terms. To discretize the ADR equations, the Crank–Nicolson finite difference method is employed with a uniform grid. The resulting nonlinear system of equations is solved using Newton’s method. At each iteration of Newton’s method, the Gauss–Seidel iterative method with sparse matrix computation is utilized to solve the block tridiagonal system and obtain the error correction vector. The consistency and stability of the numerical scheme are investigated. MATLAB codes are developed to implement this combined numerical approach. The validation of the scheme is verified by solving a two-dimensional advection-diffusion equation without reaction term. Numerical tests are provided to show the good performances of the proposed numerical scheme in simulation of ADR problems. The numerical scheme gives accurate results. The obtained numerical solutions are presented graphically. The result of this study may provide insights to apply numerical methods in solving comprehensive models of physical phenomena that capture the underlying situations.

Heavy metal transport with adsorption for instantaneous and exponential attenuation of concentration

Heavy metal pollutant is a serious problem in environmental pollution, and it is very difficult to eradicate once it enters the soil. As heavy metal adsorption has been proven to occur, the heavy metal’s behaviour can be modeled as a transport equation with adsorption. Previous adsorption term mostly due to the concentration alone, while in here, the desorption effect given by the rate of change of the concentration is also included. Also, the heavy metals are frequently considered to enter the soil after being dumped into the soil for a certain period of time. But, quick dumping onto the soil can introduce heavy metal instantaneously. Heavy metals entering the soil through leaching or when their concentration in the soil is influenced by chemical reactions, can all lead to the exponential decay of heavy metals entering the soil. Based on two-dimensional advection diffusion equation (ADE) with the new adsorption term, analytical solutions are obtained for the cases of instantaneous and exponential attenuation of heavy metals emission to soil by the method of Laplace transform. The results highlight the significant influence of emission type on the peak concentrations. If heavy metals are instantaneously enter the soil, the peak occurs in the range of 1–3 m radius from the point of emission on the first day, while for exponential attenuation the peak occurs close to the point of emission. Furthermore, there exists a correlation between retardation factors and heavy metal concentrations, where a decrease in retardation factors leads to an increase in heavy metal concentration. It is essential to investigate both types of heavy metals emission to provide valuable information for proper pollution management, effective environmental regulations and enforcement.

A modified equation analysis for immersed boundary methods based on volume penalization: Applications to linear advection–diffusion equations and high-order discontinuous Galerkin schemes

The Immersed Boundary Method (IBM) is a popular numerical approach to impose boundary conditions without relying on body-fitted grids, thus reducing the costly effort of mesh generation. To obtain enhanced accuracy, IBM can be combined with high-order methods (e.g., discontinuous Galerkin). For this combination to be effective, an analysis of the numerical errors is essential. In this work, we apply, for the first time, a modified equation analysis to the combination of IBM (based on volume penalization) and high-order methods (based on nodal discontinuous Galerkin methods) to analyze a priori numerical errors and obtain practical guidelines on the selection of IBM parameters. The analysis is performed on a linear advection–diffusion equation with Dirichlet boundary conditions. Three ways to penalize the immersed boundary are considered, the first penalizes the solution inside the IBM region (classic approach), whilst the second and third penalize the first and second derivatives of the solution. We find optimal combinations of the penalization parameters, including the first and second penalizing derivatives, resulting in minimum errors. We validate the theoretical analysis with numerical experiments for one- and two-dimensional advection–diffusion equations.

The Modified Local Boundary Knots Method for Solution of the Two-Dimensional Advection–Diffusion Equation

This paper deals with a new modification of the local boundary knots method (LBKM), which will allow the irregular node distribution and the arbitrary shape of the solution domain. Unlike previous localizations, it has no requirements on the number of nodes in the support or on the number of virtual points. Owing to the limited number of virtual points, the condition number of boundary knots matrix remains relatively low. The article contains the derivation of the relations of the method for steady and unsteady states and shows its effectiveness in three control examples.

Orthonormal piecewise Vieta-Lucas functions for the numerical solution of the one- and two-dimensional piecewise fractional Galilei invariant advection-diffusion equations

IntroductionRecently, a new family of fractional derivatives called the piecewise fractional derivatives has been introduced, arguing that for some problems, each of the classical fractional derivatives may not be able to provide an accurate statement of the consideration problem alone. In defining this kind of derivatives, several types of fractional derivatives can be used simultaneously. ObjectivesThis study introduces a new kind of piecewise fractional derivative by employing the Caputo type distributed-order fractional derivative and ABC fractional derivative. The one- and two-dimensional piecewise fractional Galilei invariant advection–diffusion equations are defined using this piecewise fractional derivative. MethodsA new class of basis functions called the orthonormal piecewise Vieta-Lucas (VL) functions are defined. Fractional derivatives of these functions in the Caputo and ABC senses are computed. These functions are utilized to construct two numerical methods for solving the introduced problems under non-local boundary conditions. The proposed methods convert solving the original problems into solving systems of algebraic equations. ResultsThe accuracy and convergence order of the proposed methods are examined by solving several examples. The obtained results are investigated, numerically. ConclusionThis study introduces a kind of piecewise fractional derivative. This derivative is employed to define the one- and two-dimensional piecewise fractional Galilei invariant advection–diffusion equations. Two numerical methods based on the orthonormal VL polynomials and orthonormal piecewise VL functions are established for these problems. The numerical results obtained from solving several examples confirm the high accuracy of the proposed methods.

Numerical Simulations of a Two-dimensional Vertically Averaged Air Pollution Measurement in a Street Canyon

Air pollution is the release of pollutants into the atmosphere that are harmful to human health and the ecosystem as a whole. Initially, urban air pollution was considered to be a regional problem caused largely by domestic heating and industrial emissions, both of which are now well under control. The building's canyon structure and the geometry of the streets in urban areas are street canyons. Side Street connects the two sides of the street, which are made up of portions of buildings. Street canyons, which are urban streets bordered on both sides by structures, have shown high levels of pollution. Pedestrians, cyclists, vehicles, and residents will most likely be surrounded by pollution concentrations higher than current air quality limits on these walkways. The research is focused on detecting air pollution in a street canyon. There will be an introduction to a transient two-dimensional advection-diffusion equation. A two-dimensional vertically averaged air pollution measurement model is utilized to characterize the air pollution concentration along a street canyon. The model delivers the pollutant concentration in the air each and every time. The model's air pollutant concentration is approximated using a finite difference technique. An approximation approach to open and closed boundary conditions is proposed. Wind direction effects are also modelled. The suggested numerical approaches performed well in producing a high level of agreement. Simple explicit schemes have the benefit of being simple to compute. These techniques may be used to simulate air pollution measurements in a variety of street canyons.

Joint Online Parameter Estimation and Optimal Sensor Placement for the Partially Observed Stochastic Advection-Diffusion Equation

In this paper, we consider the problem of jointly performing online parameter estimation and optimal sensor placement for a partially observed infinite dimensional linear diffusion process. We present a novel solution to this problem in the form of a continuous-time, two-timescale stochastic gradient descent algorithm, which recursively seeks to maximise the log-likelihood with respect to the unknown model parameters, and to minimise the expected mean squared error of the hidden state estimate with respect to the sensor locations. We also provide extensive numerical results illustrating the performance of the proposed approach in the case that the hidden signal is governed by the two-dimensional stochastic advection-diffusion equation.

The Effect of a Step Increase in Depth and Decay upon Dispersion of Coastal Effluent Discharges

Coastal wastewater-discharged effluents contain a mixture of pollutants with decay rates that vary with water depth. Analytical models using a two-dimensional advection-diffusion equation are presented to study the effects of a cross-stream sudden depth change and decay on mixing and dispersing steady discharge of effluents through a sea outfall. The solutions are illustrated graphically by plotting contours of concentration, resembling snapshots of discharged effluent plumes in the far-field. Different shapes of effluent plumes are observed due to the variability of length of the step seabed, and the concentration at the step seabed is formulated to measure how much has discharged effluents dispersed into or out of the shallow coastal waters.

A novel three-level time-split approach for solving two-dimensional nonlinear unsteady convection-diffusion-reaction equation

This paper considers a deep analysis of a three-level explicit time-split MacCormack method, namely the locally one-dimensional explicit MacCormack for the numerical solution of the two-dimensional nonlinear evolutionary advection-diffusion equation subjects to suitable initial and boundary conditions. The splitting reduces the computational cost of the algorithm. Under a suitable time-step restriction, both theoretical and numerical results on the stability and error estimates of the scheme are deeply analyzed in \(L^{m}(0,T;L^{2})\)-norm (\(m=1,2,\infty\)). The numerical experiments suggest that the proposed algorithm is easy to implement, temporal second-order convergent and fourth-order accurate in space. This shows the utility and efficiency of the considered method.

Analytical model for the two-dimensional advection-diffusion equation with the logarithmic wind profile in unstable conditions

Analytical model for the two-dimensional advection-diffusion equation with the logarithmic wind profile in unstable conditions

The Local Singular Boundary Method for Solution of Two-Dimensional Advection–Diffusion Equation

This work focuses on the derivation of the local variant of the singular boundary method (SBM) for solving the advection-diffusion equation of tracer transport. Localization is based on the combination of SBM and finite collocation. Unlike the global variant, local SBM leads to a sparse matrix of the resulting system of equations, making it much more efficient to solve large-scale tasks. It also allows solving velocity vector variable tasks, which is a problem with global SBM. This paper compares the results on several examples for the steady and unsteady variant of the advection-diffusion equation and also examines the dependence of the accuracy of the solution on the density of the nodal grid and the size of the subdomain.

A Two-Level Factored Crank-Nicolson Method for Two-Dimensional Nonstationary Advection-Diffusion Equation with Time Dependent Dispersion Coefficients and Source Terms

A Two-Level Factored Crank-Nicolson Method for Two-Dimensional Nonstationary Advection-Diffusion Equation with Time Dependent Dispersion Coefficients and Source Terms

A fourth-order two-level factored implicit scheme for solving two-dimensional unsteady transport equation with time-dependent dispersion coefficients

This paper analyzes a two-level factored implicit scheme in a numerical solution of two-dimensional unsteady advection-diffusion equation with time dependent dispersion coefficients subjects to initial and boundary conditions. The proposed approach is fast and efficient: unconditionally stable, second order accurate in time, spatial fourth order convergent and it requires less computing time. In fact, the two-level factored technique reduces to solve a tridiagonal system of linear equations at each calculation step. This reduces the computational cost of the algorithm. The analysis of the stability of the numerical scheme considers the -norm while the error estimates and convergence rate use L 2-norm. A wide set of numerical evidences are presented and discussed.

Analytical solutions to the advection-diffusion equation with Atangana-Baleanu time-fractional derivative and a concentrated loading

In this communication we have studied well known physical process of two-dimensional advection–diffusion phenomena. The advection–diffusion equation is time-fractionalized by exploiting Atangana-Baleanu fractional derivative operator. This fractionalization is achieved in the generalized constitutive equation of the mass flux density vector. The fractionalized two-dimensional advection–diffusion equation turns out to be a two-dimensional nonlinear fractional partial differential equation. This partial differential equation is considered under the hypothesis of an initial concentrated loading and Robin type boundary conditions. The analytical expression of the solution is determined for this boundary value problem by employing the integral transforms method, namely, the Laplace transform, sine-Fourier transform and finite sine–cosine Fourier transform. The effects of fractional parameter α on the concentration obtained from the analytical solution, for various parameters of interest, are illustrated graphically with the help of software Mathcad. The graphs illustrate that the memory effects are remarkable for small values of time and ordinary for large values of the time.