A Positive Finite-Difference Advection Scheme (2023)

Cited by (155)

  • How trees affect urban air quality: It depends on the source

    2022, Atmospheric Environment

    Large-eddy simulation (LES) is used to systematically analyse the impacts of trees on air quality in idealised street canyons. The LES tree model includes radiation, transpiration, drag and deposition effects. The superposition of background concentrations and local emissions is used to construct realistic urban scenarios for fine particulate matter (PM2.5) and nitrogen oxides (NOx). Both neutral and convective atmospheric conditions are considered to assess the importance of buoyancy effects and the role of tree shading and transpiration. Tree impact on local air quality is shown to be driven by the balance between the rate at which they actively remove pollutants from the air (deposition) and the way in which they alter the transport of pollutants within and out of the street canyon (dispersion). For pollutant species or street types where the concentration field is dominated by background levels (such as PM2.5), deposition will generally dominate and thus local air quality will improve. For pollutants and street types where local emission sources dominate (e.g. NOx on a busy road), the dispersion effects of trees become more prominent and can lead to elevated concentrations where mixing or exchange is significantly inhibited. Mixing in the convective simulation is more vigorous than in the neutral simulation which results in substantial differences in in-canyon flow fields and exchange velocities, highlighting the importance of incorporating thermal effects when studying urban trees. Increased residency times, and thus deposition, under neutral conditions suggest that trees can have amplified effects under conditions conducive of poor air quality. For the cases considered, trees largely act to improve air quality with the exception of localised hotspots. The competing effects of trees — specifically deposition versus altered exchange with the atmosphere — are also incorporated in a simple integral model that predicts whether or not the air quality will improve. The model matches well with LES predictions for both PM2.5 and NOx and can serve as a simple tool for urban design purposes.

  • A monotonicity-preserving higher-order accurate finite-volume method for Kapila's two-fluid flow model

    2019, Computers and Fluids

    In preparation of the study of liquefied natural gas (LNG) sloshing in ships and vehicles, we model and numerically analyze compressible two-fluid flow. We consider a five-equation two-fluid flow model, assuming velocity and pressure continuity across two-fluid interfaces, with a separate equation to track the interfaces. The system of partial differential equations is hyperbolic and quasi-conservative. It is discretized in space with a tailor-made third-order accurate finite-volume method, employing an HLLC approximate Riemann solver. The third-order accuracy is obtained through spatial reconstruction with a limiter function, for which a novel formulation is presented. The non-homogeneous term is handled in a way consistent with the HLLC treatment of the convection operator. We study the one-dimensional case of a liquid column impacting onto a gas pocket entrapped at a solid wall. It mimics the impact of a breaking wave in an LNG containment system, where a gas pocket is entrapped at the tank wall below the wave crest. Furthermore, the impact of a shock wave on a gas bubble containing the heavy gas R22, immersed in air, is simulated in two dimensions and compared with experimental results. The numerical scheme is shown to be higher-order accurate in space and capable of capturing the important characteristics of compressible two-fluid flow.

  • An extension of DG methods for hyperbolic problems to one-dimensional semi-infinite domains

    2019, Applied Mathematics and Computation

    We consider spectral discretizations of hyperbolic problems on unbounded domains using Laguerre basis functions. Taking as model problem the scalar advection equation, we perform a comprehensive stability analysis that includes strong collocation formulations, nodal and modal weak formulations, with either inflow or outflow boundary conditions, using either Gauss–Laguerre or Gauss–Laguerre–Radau quadrature and based on either scaled Laguerre functions or scaled Laguerre polynomials. We show that some of these combinations give rise to intrinsically unstable discretizations, while the combination of scaled Laguerre functions with Gauss–Laguerre–Radau quadrature appears to be stable for both strong and weak formulations. We then show how a modal discretization approach for hyperbolic systems on an unbounded domain can be naturally and seamlessly coupled to a discontinuous finite element discretization on a finite domain. An example of one dimensional hyperbolic system is solved with the proposed domain decomposition technique. The errors obtained with the proposed approach are found to be small, enabling the use of the coupled scheme for the simulation of Rayleigh damping layers in the semi-infinite part. Energy errors and reflection ratios of the scheme in absorbing wavetrains and single Gaussian signals show that a small number of modes in the semi-infinite domain are sufficient to damp the waves. The theoretical insight and numerical results corroborate previous findings by the authors and establish the scaled Laguerre functions-based discretization as a flexible and efficient tool for absorbing layers as well as for the accurate simulation of waves in unbounded regions.

  • Positivity-preserving scheme for two-dimensional advection–diffusion equations including mixed derivatives

    2018, Computer Physics Communications

    Citation Excerpt :

    Section 3 discusses the Fokker–Planck collision operator in cylindrical coordinates, including the assumption of local thermal equilibrium and thermal equilibration tests, and is followed by a short summary in Section 4. Positivity-preserving schemes to two-dimensional advection–diffusion equations have been studied in detail [2,3], but solving problems with mixed derivative terms have received much less attention. The reason for this is that typically a change of coordinate system can be performed in order to eliminate the mixed derivative terms, or the mixed derivative terms are weak compared to the advection–diffusion terms and can therefore be neglected.

    (Video) Finite Difference Schemes for Advection and Diffusion

    In this work, we propose a positivity-preserving scheme for solving two-dimensional advection–diffusion equations including mixed derivative terms, in order to improve the accuracy of lower-order methods. The solution of these equations, in the absence of mixed derivatives, has been studied in detail, while positivity-preserving schemes for mixed derivative terms have received much less attention. A two-dimensional diffusion equation, for which the analytical solution is known, is solved numerically to show the applicability of the scheme. It is further applied to the Fokker–Planck collision operator in two-dimensional cylindrical coordinates under the assumption of local thermal equilibrium. For a thermal equilibration problem, it is shown that the scheme conserves particle number and energy, while the preservation of positivity is ensured and the steady-state solution is the Maxwellian distribution.

  • MPDATA: Third-order accuracy for variable flows

    2018, Journal of Computational Physics

    This paper extends the multidimensional positive definite advection transport algorithm (MPDATA) to third-order accuracy for temporally and spatially varying flows. This is accomplished by identifying the leading truncation error of the standard second-order MPDATA, performing the Cauchy–Kowalevski procedure to express it in a spatial form and compensating its discrete representation—much in the same way as the standard MPDATA corrects the first-order accurate upwind scheme. The procedure of deriving the spatial form of the truncation error was automated using a computer algebra system. This enables various options in MPDATA to be included straightforwardly in the third-order scheme, thereby minimising the implementation effort in existing code bases. Following the spirit of MPDATA, the error is compensated using the upwind scheme resulting in a sign-preserving algorithm, and the entire scheme can be formulated using only two upwind passes. Established MPDATA enhancements, such as formulation in generalised curvilinear coordinates, the nonoscillatory option or the infinite-gauge variant, carry over to the fully third-order accurate scheme. A manufactured 3D analytic solution is used to verify the theoretical development and its numerical implementation, whereas global tracer-transport benchmarks demonstrate benefits for chemistry-transport models fundamental to air quality monitoring, forecasting and control. A series of explicitly-inviscid implicit large-eddy simulations of a convective boundary layer and explicitly-viscid simulations of a double shear layer illustrate advantages of the fully third-order-accurate MPDATA for fluid dynamics applications.

  • A lattice method for the Eulerian simulation of heavy particle suspensions

    2016, Comptes Rendus - Mecanique

    Modeling dispersed solid phases in fluids still represents a computational challenge when considering a small-scale coupling in wide systems, such as the atmosphere or industrial processes at high Reynolds numbers. A numerical method is here introduced for simulating the dynamics of diffusive heavy inertial particles in turbulent flows. The approach is based on the position/velocity phase–space particle distribution. The discretization of velocities is inspired from lattice Boltzmann methods and is chosen to match discrete displacements between two time steps. For each spatial position, the time evolution of particles momentum is approximated by a finite-volume approach. The proposed method is tested for particles experiencing a Stokes viscous drag with a prescribed fluid velocity field in one dimension using a random flow, and in two dimensions with the solution to the forced incompressible Navier–Stokes equations. Results show good agreements between Lagrangian and Eulerian dynamics for both spatial clustering and the dispersion in particle velocities. In particular, the proposed method, in contrast to hydrodynamical Eulerian descriptions of the dispersed phase, is able to reproduce fine particle kinetic phenomena, such as caustic formation or trajectory crossings. This indicates the suitability of this approach at large Stokes numbers for situations where details of collision processes are important.

    La modélisation de particules solides dispersées dans un fluide reste actuellement un défi numérique, surtout lorsqu'il y a une grande séparation d'échelles entre le couplage et l'écoulement, comme par exemple dans l'atmosphère ou les écoulements industriels à grand nombre de Reynolds. Une méthode numérique est ici présentée dans le but de simuler la dynamique de particules lourdes diffusives dans des écoulements turbulents. L'approche est basée sur la distribution des particules dans l'espace des phases positions–vitesses. La discrétisation en vitesses s'inspire de la méthode de Boltzmann sur réseau et est choisie de telle manière à ce qu'elle corresponde à des déplacements discrets entre deux pas de temps. Pour chaque position spatiale, l'évolution temporelle de la quantité de mouvement est résolue par une approche de volumes finis. La méthode proposée est testée pour des particules soumises à un frottement visqueux de Stokes, avec des écoulements aléatoires en une dimension, et avec des solutions de l'équation de Navier–Stokes incompressible forcée en deux dimensions. Les résultats montrent un bon accord entre les simulations lagrangiennes et eulériennes pour reproduire les concentrations préférentielles et la dispersion des vitesses des particules. De plus, la méthode proposée, contrairement à des descriptions hydrodynamiques des suspensions, permet de résoudre le croisement de trajectoires et la formation de caustiques, ce qui montre sa pertinence aux grands nombres de Stokes pour les situations où les détails des processus de collision sont importants.

    (Video) Explicit Upwind Finite Difference Solution to the Advection Equation
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What is advection scheme? ›

In computational physics, the term upwind scheme (sometimes advection scheme) typically refers to a class of numerical discretization methods for solving hyperbolic partial differential equations, in which so-called upstream variables are used to calculate the derivatives in a flow field.

What is the equation for the finite difference method of transport? ›

τh(h,∆t) = O(hp) + O(∆tq).

What is an example of advection? ›

An example of advection is the transport of pollutants or silt in a river by bulk water flow downstream. Another commonly advected quantity is energy or enthalpy. Here the fluid may be any material that contains thermal energy, such as water or air.

Which is an example of advection quizlet? ›

Advection is the horizontal transfer of energy. An example is wind blowing across the globe or energy going from cloud to cloud.

What is finite difference method for advection? ›

General form of the 1D Advection-Diffusion Problem The general form of the 1D advection-diffusion is given as: dU dt = ϵ d2U dx2 − a dU dx + F (1) where, U is the variable of interest t is time ϵ is the diffusion coefficient a is the average velocity F describes ”sources” or ”sinks” of the quantity U.

What is the finite difference method? ›

The finite difference method is directly applied to the differential form of the governing equations. The principle is to employ a Taylor series expansion for the discretisation of the derivatives of the flow variables.

What is finite difference method method? ›

The finite difference method (FDM) is an approximate method for solving partial differential equations. It has been used to solve a wide range of problems. These include linear and non-linear, time independent and dependent problems.

Can advection be negative? ›

There are essentially two types of advection: positive and negative.

What is the process of advection? ›

Advection is the process by which microbes are carried by the bulk motion of the flowing groundwater. As long as they do not interact with the surface of soil grains, microorganisms are transported through the porous medium by advection at an average rate equal to the average velocity of the water.

How does advection occur? ›

Very simply, advection occurs any time an airmass moves. When a warm airmass moves into an area previously occupied by a cooler airmass, Warm Air Advection (WAA) occurs. Cold air replacing warm air is known as Cold Air Advection (CAA). Each of these processes unfolds differently, and produces different results.

What is an example of advection and diffusion? ›

Mixing of a drop of ink (chemicals) in a flowing river is both diffusion and advection. Think about tornadoes, anything that falls in the vortex of tornado tears down (diffusion), mixes up in the circular motion (advection) and carries away with the tornado (movement of the vortex) (convection).

Does advection transfer heat? ›

The transfer of heat through the horizontal movement of air is called advection. The horizontal movement of the air is comparatively more significant than the vertical movement. Most of the diurnal variation in weather is caused by advection only in the middle latitudes.

What are the different types of air advection? ›

While the most common examples are warm air or cold air advection, the term can be used to describe other horizontal transport processes such as moisture advection, where atmospheric water vapor is transported by wind. Occasionally, warm or cold air advection is described simply as "temperature advection".

What are the three types of finite differences? ›

Three basic types are commonly considered: forward, backward, and central finite differences.

How is finite difference method used in heat transfer? ›

The finite difference method is one way to solve the governing partial differential equations into numerical solutions in a heat transfer system. This is done through approximation, which replaces the partial derivatives with finite differences. This provides the value at each grid point in the domain.

What is finite difference method with convection? ›

In a finite difference formulation, the spatial oscillations are reduced by a family of discretization schemes like upwind scheme. In this method, the basic shape function is modified to obtain the upwinding effect. This method is an extension of Runge–Kutta discontinuous for a convection-diffusion equation.

What is the advantage of finite difference method? ›

The numerical method of solving differential equations by approximating them with difference equations is called the finite difference method.
The Finite Difference Method.
Gives higher-order accuracy of the spatial discretization.The finite difference method requires a structured grid.
2 more rows

What is finite difference method for two variables? ›

The finite difference procedure evaluates the dependent variables (pressure and saturation) at discrete points in space and in time. The derivatives are approximated by a difference of the dependent variable between two or more discrete points in space or in time.

What is stability of finite difference method? ›

A finite difference scheme is stable if the errors made at one time step of the calculation do not cause the errors to be magnified as the computations are continued. A neutrally stable scheme is one in which errors remain constant as the computations are carried forward.

What is the difference between fem and FDM? ›

in the FDM methods, the discretisation of the domain is done as a set of nodes at which the results are determined, while in the FEM method the results are known in every point of the domain as the approximation is done with functions defined on small triangular (or quadrilateral) areas in 2D.

What is the order of accuracy of the finite difference scheme? ›

Definition: The power of Δx with which the truncation error tends to zero is called the Order of Accuracy of the Finite Difference approximation. The Taylor Series Expansions: FD and BD are both first order or are O(Δx) (Big-O Notation) CD is second order or are O(Δx2) (Big-O Notation)

What is advection in the climate system? ›

Very simply, advection occurs any time an airmass moves. When a warm airmass moves into an area previously occupied by a cooler airmass, Warm Air Advection (WAA) occurs. Cold air replacing warm air is known as Cold Air Advection (CAA). Each of these processes unfolds differently, and produces different results.

What is advection in geotechnical engineering? ›

Advection: It is the process by which the contaminants are transported by overall motion of flowing groundwater.

What is the difference between convention and advection? ›

Convection is the movement of a fluid, typically in response to heat. Advection is the movement of some material dissolved or suspended in the fluid. So if you have pure water and you heat it you will get convection of the water.

What does advection mean in hydrological cycle? ›

In the water cycle, advection is the movement of water through the atmosphere. It's the process that allows water from the ocean to be recycled and make its way to our drinking fountains. It also allows water that was once deep underground to be recycled as snow on top of mountains.

Is cold air advection positive or negative? ›

Temperature advection is measured as a change in temperature per unit time, and the common units on temperature advection are degrees Fahrenheit per hour, with positive values indicating warm-air advection and negative values representing cold-air advection.

What are the different types of advection? ›

There are essentially two types of advection: positive and negative. Figure D below shows positive advection with higher values of a variable (in this case temperature) being advected towards lower values. The end result of positive advection is to increase the variable values in the direction the wind is blowing.

What is an example of advection in the atmosphere? ›

advection, in atmospheric science, change in a property of a moving mass of air because the mass is transported by the wind to a region where the property has a different value (e.g., the change in temperature when a warm air mass moves into a cool region).

How does advection affect the climate? ›

Put in another way, if there is warm advection occuring at a given station, expect the temperatures to increase. In contrast, if cold advection is occurring at a given station, expect the temperatures to drop. Temperature advection refers to change in temperature caused by movement of air by the wind.

How do you determine advection? ›

If isotherms are approaching your point of interest that are colder than the temperature at your point of interest, then it is cold air advection. If the isotherms are warmer, then it is warm air advection.

Why is advection more important than convection? ›

In short, advection is specifically dependent on the currents within a fluid, while convection is a broader process that also contains diffusive heat transfer. As such, advection can only exist in gases and liquids, and it isn't possible in a solid.

Is advection better than diffusion? ›

So, advection increases the concentration gradients in the fluid whereas diffusion wipe out these gradients. Notice that advection is crucial in order to achive mixing (homogenisation of cream/sugar). In the absence of advection, diffusion will take months to mix your coffee (that will be boring).

What is advection in geology? ›

Advection is a lateral or horizontal transfer of mass, heat, or other property. Accordingly, winds that blow across Earth's surface represent advectional movements of air. Advection also takes place in the ocean in the form of currents.

What is advective vs diffusive flow? ›

Advection refers to the transport mechanism of a substance (or conserved property i.e. mass) by a fluid due to the fluid's bulk motion (i.e. to the flow). Diffusion refers to another transport mechanism which occurs without any motion of the fluid´s bulk.

What is advection also known as? ›

The term advection sometimes serves as a synonym for convection, but technically, convection covers the sum of transport both by diffusion and by advection.


1. 6. The Advection Diffusion Equation (numerical solution with FDM)
(Professor Saad Explains)
2. Higher order schemes to Finite Difference Method
(Vipul Patel)
3. Finite-Volume Discretization of Advection-Diffusion Equation
(Sandip Mazumder)
4. Lecture 5 - Part a: FVM for linear advection eq. (cont.)
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5. Finite difference for advection hyerbolic equation
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6. Solution error from central difference scheme for advection equation
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