Time-dependent Advection-diffusion Equations Research Articles

A fast solvable operator-splitting scheme for time-dependent advection diffusion equation

It is well known that the implicit central difference discretization for unsteady advection diffusion equation (ADE) suffers from being time-consuming to solve when the advection term dominates. In this paper, we propose an operator-splitting scheme for the unsteady ADE, in which the ADE is firstly discretized by Crank-Nicolson (CN) scheme in time and central difference scheme in space; and then the discrete advection-diffusion problem is split as advection sub-problem and diffusion sub-problem at each time-level. The significance of the new scheme is that these sub-problems can be fast and directly solved within a linearithmic complexity (a linear-times-logarithm complexity) by means of fast sine transforms (FSTs). In particular, the complexity is independent of the dominance of the advection term. Theoretically, we show that proposed scheme is unconditionally stable and of second-order convergence in time and space. Numerical results are reported to show the efficiency of the proposed scheme.

Fast Models for Predicting Pollutant Dispersion inside Urban Canopies

A fast pollutant dispersion model for urban canopies is developed by coupling mean wind profiles to a parameterisation of turbulent diffusion and solving the time-dependent advection–diffusion equation. The performance of a simplified, coarse-grained representation of the velocity field is investigated. Spatially averaged mean wind profiles within local averaging regions or repeating units are predicted by solving the three-dimensional Poisson equation for a set of discrete vortex sheets. For each averaging region, the turbulent diffusion is parameterised in terms of the mean wind profile using empirical constants derived from large-eddy simulation (LES). Nearly identical results are obtained whether the turbulent fluctuations are specified explicitly or an effective diffusivity is used in their place: either version of the fast dispersion model shows much better agreement with LES than does the Gaussian plume model (e.g., the normalized mean square error inside the canopy is several times smaller). Passive scalar statistics for a regular cubic building array show improved agreement with LES when wind profiles vary in the horizontal. The current implementation is around 50 times faster than LES. With its combination of computational efficiency and moderate accuracy, the fast model may be suitable for time-critical applications such as emergency dispersion modelling.

Moment analysis for predicting effective transport properties in hierarchical retentive porous media

We report on a new homogenization approach to solve, with drastically improved speed and accuracy, the general advection-diffusion equation in hierarchical porous media with localized diffusion and adsorption/desorption processes, thus opening the way to a much deeper understanding of the band broadening process in chromatographic systems. The proposed robust and efficient moment-based approach allows us to compute the exact local and integral concentration moments and hence provides exact solutions for the effective velocity and dispersion coefficients of migrating solute particles. Innovative to the proposed method is also that it not only produces the exact effective transport parameters of the long-time asymptotic solution, but also their entire transient. The analysis of the transient behaviour can be used, for example, to properly identify the time and length scales needed to achieve the macro-transport conditions. If the hierarchical porous media can be represented as the periodic repetition of a unit lattice cell, the method only requires the solution of the time-dependent advection-diffusion equations for the zeroth order and first-order exact local moments, exclusively on the unit cell. This implies an enormous reduction of the computational efforts and a significant improvement of the accuracy of the results when compared to the direct numerical simulation (DNS) approaches which require flow domains that are long enough to achieve steady-state conditions, and hence often cover tens to hundreds of unit cells. The reliability of the proposed method is verified by comparing its predictions with DNS results, in one, two and three dimensions, in both transient and asymptotic conditions. The influence of top and bottom no-slip walls on the separation performance of chromatographic columns with micromachined porous and nonporous pillars is discussed in detail.

HIFIR: Hybrid Incomplete Factorization with Iterative Refinement for Preconditioning Ill-Conditioned and Singular Systems

We introduce a software package called Hybrid Incomplete Factorization with Iterative Refinement (HIFIR) for preconditioning sparse, unsymmetric, ill-conditioned, and potentially singular systems. HIFIR computes a hybrid incomplete factorization (HIF) , which combines multilevel incomplete LU factorization with a truncated, rank-revealing QR (RRQR) factorization on the final Schur complement. This novel hybridization is based on the new theory of ϵ-accurate approximate generalized inverse (AGI) . It enables near-optimal preconditioners for consistent systems and enables flexible GMRES to solve inconsistent systems when coupled with iterative refinement. In this article, we focus on some practical algorithmic and software issues of HIFIR. In particular, we introduce a new inverse-based rook pivoting (IBRP) into ILU, which improves the robustness and the overall efficiency for some ill-conditioned systems by significantly reducing the size of the final Schur complement for some systems. We also describe the software design of HIFIR in terms of its efficient data structures for supporting rook pivoting in a multilevel setting, its template-based generic programming interfaces for mixed-precision real and complex values in C++, and its user-friendly high-level interfaces in MATLAB and Python. We demonstrate the effectiveness of HIFIR for ill-conditioned or singular systems arising from several applications, including the Helmholtz equation, linear elasticity, stationary incompressible Navier–Stokes (INS) equations, and time-dependent advection-diffusion equation.

Three-dimensional simulations of the airborne COVID-19 pathogens using the advection-diffusion model and alternating-directions implicit solver

In times of the COVID-19, reliable tools to simulate the airborne pathogens causing the infection are extremely important to enable the testing of various preventive methods. Advection-diffusion simulations can model the propagation of pathogens in the air. We can represent the concentration of pathogens in the air by “contamination” propagating from the source, by the mechanisms of advection (representing air movement) and diffusion (representing the spontaneous propagation of pathogen particles in the air). The three-dimensional time-dependent advection-diffusion equation is difficult to simulate due to the high computational cost and instabilities of the numerical methods. In this paper, we present alternating directions implicit isogeometric analysis simulations of the three-dimensional advection-diffusion equations. We introduce three intermediate time steps, where in the differential operator, we separate the derivatives concerning particular spatial directions. We provide a mathematical analysis of the numerical stability of the method. We show well-posedness of each time step formulation, under the assumption of a particular time step size. We utilize the tensor products of one-dimensional B-spline basis functions over the three-dimensional cube shape domain for the spatial discretization. The alternating direction solver is implemented in C++ and parallelized using the GALOIS framework for multi-core processors. We run the simulations within 120 minutes on a laptop equipped with i7 6700 Q processor 2.6 GHz (8 cores with HT) and 16 GB of RAM. © 2021 The Author(s).

Solving Advection-Diffusion Equations via Sobolev Space Notions

In this paper, the time-dependent advection-diffusion equation is studied. After introducing these equations in various engineering fields such as gas adsorption, solid dissolution, heat and mass transfer in falling film or pipe and other equations similar to transport phenomena, a new method has been proposed to find their solutions. Among the various works on solving these PDEs by numerical and somewhat analytical methods, a general analytical framework for solving these equations is presented. Using advanced components of Sobolev spaces, weak solutions and some important integral inequalities, an analytical method for the existence and uniqueness of the weak solution of these PDEs is presented, which is the best solution in the proposed structure. Then, with a reduced system of ODE, one can solve the problem of the general parabolic boundary value problem, which includes PDE transport phenomena. Besides, the new approach supports the infinite propagation speed of disturbances of (time-dependent) diffusion-time equations in semi-infinite media.

Two-Grid Based Adaptive Proper Orthogonal Decomposition Method for Time Dependent Partial Differential Equations

In this article, we propose a two-grid based adaptive proper orthogonal decomposition (POD) method to solve the time dependent partial differential equations. Based on the error obtained in the coarse grid, we propose an error indicator for the numerical solution obtained in the fine grid. Our new method is cheap and easy to be implement. We apply our new method to the solution of time-dependent advection–diffusion equations with the Kolmogorov flow and the ABC flow. The numerical results show that our method is more efficient than the existing POD methods.

Finite element simulation of physical systems in augmented reality

We present a system for immersive and interactive physics simulation in augmented reality, implemented as an application on the Microsoft HoloLens. The system allows a user to simulate physical phenomena in real-world geometries by scanning the surroundings and automatically creating a finite element discretization (mesh). The system also provides a user interface based on voice commands for specification of boundary conditions and source terms, and for immersive visualization of simulation results. All computations are carried out on the HoloLens. The current implementation, named HoloFEM, demonstrates the application of augmented reality simulation for the time-dependent advection–diffusion equation. In the current work, we consider a simplified model problem with simplified parameters and data. The model is however still conceptually applicable to the simulation of indoor climate (temperature and air quality), which has implications for interactive HVAC (heating, ventilation, and air conditioning) simulation.

A solution of the time-dependent advection-diffusion equation

A solution of the time-dependent advection-diffusion equation

A solution of the time-dependent advection-diffusion equation

We present an analytical solution that considers time dependence in the wind profile and the eddy diffusivity. The solution accepts any profiles of wind and eddy coefficient diffusions in a limited planetary boundary layer. The solution is based on the idea of a decomposition method upon expanding the pollutant concentration in a truncated series. This produces a set of recursive equations whose solutions are known. Each equation of this set is solved by the generalised integral Laplace transform technique (GILTT) method. The solution's ability to represent real situations was checked, comparing model predictions with the over-land along-wind dispersion (OLAD) experimental dataset.

Explicit and implicit forms of differential quadrature method for advection–diffusion equation with variable coefficients in semi-infinite domain

In this paper, a numerical solution of one-dimensional time-dependent advection-diffusion equation with variable coefficients in semi-infinite domain is presented by using the differential quadrature method. Both the explicit and implicit approaches are provided. Totally, two solute dispersion problems are employed to simulate various conditions. The inhomogeneity of the domain is supplied by the spatially dependent flow. The problem domains are modeled with Chebyshev–Gauss–Lobatto grid points. In order to examine the accuracy and the efficiency of the suggested explicit and implicit approaches, analytical solutions, which are presented in the literature, are employed. In addition, the results of the above-mentioned method are compared with outcomes of the finite difference method. The results show that both of the explicit and implicit forms of the differential quadrature method are efficient, robust and reliable. But between these two forms, numerical predictions of implicit form are more accurate than explicit form.

Approximation of stochastic advection diffusion equations with Stochastic Alternating Direction Explicit methods

The numerical solutions of stochastic partial differential equations of Itô type with time white noise process, using stable stochastic explicit finite difference methods are considered in the paper. Basically, Stochastic Alternating Direction Explicit (SADE) finite difference schemes for solving stochastic time dependent advection-diffusion and diffusion equations are represented and the main properties of these stochastic numerical methods, e.g. stability, consistency and convergence are analyzed. In particular, it is proved that when stable alternating direction explicit schemes for solving linear parabolic PDEs are developed to the stochastic case, they retain their unconditional stability properties applying to stochastic advection-diffusion and diffusion SPDEs. Numerically, unconditional stable SADE techniques are significant for approximating the solutions of the proposed SPDEs because they do not impose any restrictions for refining the computational domains. The performance of the proposed methods is tested for stochastic diffusion and advection-diffusion problems, and the accuracy and efficiency of the numerical methods are demonstrated.

Bubble and multiscale stabilization of bilinear finite element methods for transient advection–diffusion equations on rectangular grids

In this paper a Petrov–Galerkin type stabilization for a time dependent advection–diffusion equation is considered. We first enrich the bilinear test space with bubble functions and the bilinear trial space with a special combination of bubble and multiscale functions for the steady state advection–diffusion equation. It is known that solving the residual equation obtained by the bubble elimination procedure is as difficult as solving the steady case of the original problem, which makes the enriched methods quite costly for two-dimensional problems. In this study, we instead utilize their cheap and efficient approximations in each rectangular element. Then we suggest a recipe for a proper adaptation of these functions combined with the generalized Euler time integration to the unsteady problem. Some numerical experiments for typical model problems are presented to illustrate the accuracy of our method.

Stellar evolution of massive stars with a radiative α-Ω dynamo

Models of rotationally-driven dynamos in stellar radiative zones have suggested that magnetohydrodynamic transport of angular momentum and chemical composition can dominate over the otherwise purely hydrodynamic processes. A proper consideration of the interaction between rotation and magnetic fields is therefore essential. Previous studies have focused on a magnetic model where the magnetic field strength is derived as a function of the stellar structure and angular momentum distribution. We have adapted our one-dimensional stellar rotation code, RoSE, to model the poloidal and toroidal magnetic field strengths with a pair of time-dependent advection-diffusion equations coupled to the equations for the evolution of the angular momentum distribution and stellar structure. This produces a much more complete, though still reasonably simple, model for the magnetic field evolution. Our model reproduces well observed surface nitrogen enrichment of massive stars in the Large Magellanic Cloud. In particular it reproduces a population of slowly-rotating nitrogen-enriched stars that cannot be explained by rotational mixing alone alongside the traditional rotationlly-enriched stars. The model further predicts a strong mass-dependency for the dynamo-driven field. Above a threshold mass, the strength of the magnetic dynamo decreases abruptly and so we predict that more massive stars are much less likely to support a dynamo-driven field than less massive stars.

Error estimates for triangular and tetrahedral finite elements in combination with a trajectory approximation of the first derivatives for advection-diffusion equations

In this paper, a modified method of characteristics in combination with integral identities of triangular and tetrahedral linear elements is used to prove a uniform optimal-order error estimate that depends only on the initial data and right-hand side, but not on a scaling parameter ɛ, for multidimensional time-dependent advection-diffusion equations.

Numerical analysis of a least-squares finite element method for the time-dependent advection–diffusion equation

A mixed finite element scheme designed for solving the time-dependent advection–diffusion equations expressed in terms of both the primal unknown and its flux, incorporating or not a reaction term, is studied. Once a time discretization of the Crank–Nicholson type is performed, the resulting system of equations allows for a stable approximation of both fields, by means of classical Lagrange continuous piecewise polynomial functions of arbitrary degree, in any space dimension. Convergence in the norm of H 1 × H ( div ) in space and in appropriate senses in time applying to this pair of fields is demonstrated.

Puff Models for Simulation of Fugitive Hazardous Emissions in Atmosphere

A puff model for the dispersion of material from fugitive hazardous emissions is presented. For vertical diffusion the model is based on general techniques for solving time dependent advection-diffusion equation: the ADMM (Advection Diffusion Multilayer Method) and GILTT (Generalized Integral Laplace Transform Technique) techniques. Both approaches accept wind and eddy diffusion coefficients with any restriction in their height functions. Comparisons between values predicted by the models against experimental ground-level concentrations (from Copenhagen data set) are shown. The preliminary results confirm the applicability of the approaches proposed and are promising for future work.

A multilayer model to simulate rocket exhaust clouds

This paper presents the MSDEF (Modelo Simulador da Dispersao de Efluentes de Foguetes, in Portuguese) model, which represents the solution for time-dependent advection-diffusion equation applying the Laplace transform considering the Atmospheric Boundary Layer as a multilayer system. This solution allows a time evolution description of the concentration field emitted from a source during a release lasting time t r , and it takes into account deposition velocity, first-order chemical reaction, gravitational settling, precipitation scavenging, and plume rise effect. This solution is suitable for describing critical events relative to accidental release of toxic, flammable, or explosive substances. A qualitative evaluation of the model to simulate rocket exhaust clouds is showed.

Domain decomposition and model reduction for the numerical solution of PDE constrained optimization problems with localized optimization variables

We introduce a technique for the dimension reduction of a class of PDE constrained optimization problems governed by linear time dependent advection diffusion equations for which the optimization variables are related to spatially localized quantities. Our approach uses domain decomposition applied to the optimality system to isolate the subsystem that explicitly depends on the optimization variables from the remaining linear optimality subsystem. We apply balanced truncation model reduction to the linear optimality subsystem. The resulting coupled reduced optimality system can be interpreted as the optimality system of a reduced optimization problem. We derive estimates for the error between the solution of the original optimization problem and the solution of the reduced problem. The approach is demonstrated numerically on an optimal control problem and on a shape optimization problem.

A uniform estimate for the MMOC for two‐dimensional advection‐diffusion equations

AbstractWe prove an optimal‐order error estimate in a weighted energy norm for the modified method of characteristics (MMOC) and the modified method of characteristics with adjusted advection (MMOCAA) for two‐dimensional time‐dependent advection‐diffusion equations, in the sense that the generic constants in the estimates depend on certain Sobolev norms of the true solution but not on the scaling diffusion parameter ε. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010