Unsteady Advection Research Articles

A Surrogate Reduced Order Model of the Unsteady Advection Dominant Problems Based on Combination of Deep Autoencoders-LSTM and POD

A Surrogate Reduced Order Model of the Unsteady Advection Dominant Problems Based on Combination of Deep Autoencoders-LSTM and POD

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.

A digital twin model for contaminant fate and transport in urban and natural drainage networks with online state estimation

Increased pollutant loads caused by urbanization and climate change have led to widespread impairment of surface water systems. To better manage these threats, water managers are seeking digital twins that combine online models with sensor data to respond to water quality hazards in real-time. This study introduces Pipedream-WQ, a new model for contaminant transport in drainage networks that combines a novel implicit solver for the unsteady advection–reaction–diffusion (ARD) equation with an efficient data assimilation scheme based on Kalman Filtering. We show that this solver reliably reproduces analytical solutions to the ARD equation in steady conditions, and accurately captures unsteady contaminant transport behavior in a complex drainage network. Furthermore, we show that online sensor data assimilation enables better estimation of contaminant concentrations at ungauged locations compared to a model-only approach. This model will enable improved pollutant tracking and source identification, and active water quality management through real-time control of hydraulic infrastructure.

Deep learning solver for solving advection–diffusion​ equation in comparison to finite difference methods

In numerical modeling, the advection–diffusion equation describes the long-range transport of atmospheric pollutants. Most numerical models in the atmospheric science community are based on finite difference methods (FDM). In this study, we conduct a comprehensive comparative analysis of standard FDM-based numerical solvers with a deep learning-based solver, the objective of which is to solve the 2D unsteady advection–diffusion equation. The performance is compared on key performance aspects accuracy, stability, and interpolation. In the analysis, we find that despite being trained with a coarse resolution, the DNN solver is the most accurate among all the solvers. For the DNN solver, the mean absolute error and maximum absolute error of fluid concentration are lowered up to 2 orders of magnitude than the FDM-based method, which corresponds to 95% and 97% relative error reduction, respectively. The analysis also shows that the DNN solver is more stable in coarse spatial–temporal domains. Owing to its continuous nature, the DNN can interpolate a solution with consistent accuracy in a resampled spatial and temporal domain magnified up to 5 and 16 times, respectively. This study highlights the fundamental differences in the partial differential equation solving methods by comparing the DNN and FDM-based solvers and presents the DNN solver as a potential alternative to the FDM-based solvers in atmospheric numerical modeling.

A cubic B-spline quasi-interpolation method for solving two-dimensional unsteady advection diffusion equations

PurposeThe purpose of this study is to extend the cubic B-spline quasi-interpolation (CBSQI) method via Kronecker product for solving 2D unsteady advection-diffusion equation. The CBSQI method has been used for solving 1D problems in literature so far. This study seeks to use the idea of a Kronecker product to extend the method for 2D problems.Design/methodology/approachIn this work, a CBSQI is used to approximate the spatial partial derivatives of the dependent variable. The idea of the Kronecker product is used to extend the method for 2D problems. This produces the system of ordinary differential equations (ODE) with initial conditions. The obtained system of ODE is solved by strong stability preserving the Runge–Kutta method (SSP-RK-43).FindingsIt is found that solutions obtained by the proposed method are in good agreement with the analytical solution. Further, the results are also compared with available numerical results in the literature, and a reasonable degree of compliance is observed.Originality/valueTo the best of the authors’ knowledge, the CBSQI method is used for the first time for solving 2D problems and can be extended for higher-dimensional problems.

High-order continuous Galerkin methods for multi-dimensional advection–reaction–diffusion problems

We construct in this paper two efficient spectral algorithms in the frequency space for solving unsteady advection–reaction–diffusion equations with constant and variable coefficients. We first consider a Jacobi–Galerkin method for solving linear equations with constant coefficients. We then develop the direct solution algorithm for the linear advection–reaction–diffusion equations with variable coefficients using the Jacobi–Galerkin method with numerical integration. The proposed Jacobi–Galerkin methods, both in temporal and spatial discretizations, are successfully developed to handle the two-dimensional unsteady advection–reaction–diffusion equations with constant and variable coefficients and with fractional orders. In these methods, the model solution is expanded in both space and time in terms of polynomials bases built upon a linear combination of Jacobi polynomials. The homogeneous initial and Dirichlet boundary conditions are satisfied exactly by expanding the model solution in terms of these polynomials. The proposed Jacobi–Galerkin methods yield an exponential rate of convergence when the solution is smooth and allow a great flexibility to handle multi-dimensional time fractional advection–reaction–diffusion equations. Finally, a series of numerical examples are presented to demonstrate the efficiency and flexibility of the methods.

An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

PurposeThis paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method.Design/methodology/approachA bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids.FindingsBi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also.Originality/valueADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

A high-order discontinuous Galerkin method for unsteady advection–diffusion problems

A high-order discontinuous Galerkin method with Lagrange multipliers is presented for the solution of unsteady advection–diffusion problems in the high Péclet number regime. It operates directly on the second-order form of the governing equation and does not require any stabilization. Its spatial basis functions are chosen among the free-space solutions of the homogeneous form of the partial differential equation obtained after time-discretization. It also features Lagrange multipliers for enforcing a weak continuity of the approximated solution across the element interface boundaries. This leads to a system of differential–algebraic equations which are time-integrated by an implicit family of schemes. The numerical stability of these schemes and the well-posedness of the overall discretization method are supported by a theoretical analysis. The performance of this method is demonstrated for various high Péclet number constant-coefficient model flow problems.

Numerical solution of unsteady advection dispersion equation arising in contaminant transport through porous media using neural networks

A soft computing approach based on artificial neural network (ANN) and optimization is presented for the numerical solution of the unsteady one-dimensional advection–dispersion equation (ADE) arising in contaminant transport through porous media. A length factor ANN method, based on automatic satisfaction of arbitrary boundary conditions (BCs) was chosen for the numerical solution of ADE. The strength of ANN is exploited to construct a trial approximate solution (TAS) for ADE in a way that it satisfies the initial or BCs exactly. An unsupervised error is constructed in approximating the solution of ADE which is minimized by training ANN using gradient descent algorithm (GDA). Two challenging test problems of ADE are considered in this paper, in which, the first problem has steep boundary layers near x=1 and many numerical methods create non-physical oscillation near steep boundaries. Also for the second problem many numerical schemes suffer from computational noise and instability issues. The proposed method is advantageous as it does not require temporal discretization for the solution of the ADEs as well as it does not suffer from numerical instability. The reliability and effectiveness of the presented algorithm is investigated by sufficient large number of independent runs and comparison of results with other existing numerical methods. The results show that the present method removes the difficulties arising in the solution of the ADEs and provides solution with good accuracy.

A Lagrangian–Eulerian finite element algorithm for advection–diffusion–reaction problems with phase change

This paper presents a particle-based Lagrangian–Eulerian algorithm for the solution of the unsteady advection–diffusion–reaction heat transfer equation with phase change. The algorithm combines a Lagrangian formulation for the advection + reaction problem with the Eulerian-based heat source method for the diffusion + phase change problem. The coupling between the Lagrangian and Eulerian subproblems is achieved with a phase change detector scheme based on a local latent heat balance and a consistent/conservative interpolation technique between Lagrangian particles and the Eulerian grid. This technique makes use of an auxiliary (finer) Eulerian grid that provides a simple and efficient method of tracking internal heterogeneities (e.g. phase boundaries), allows the use of higher order integration quadratures, and facilitates the implementation of multiscale techniques. The performance of the proposed algorithm is compared against one- and two-dimensional benchmark problems, i.e. pure rigid-body advection, isothermal and non-isothermal phase change, two-phase advective heat transfer and chemical reactions coupled with diffusion and advection. The numerical results confirm that the proposed solution method is accurate, oscillation-free and useful for and applicable to a wide range of fully coupled problems in science and engineering.

An optimal time-stepping algorithm for unsteady advection–diffusion problems

We consider the numerical solution of a linear time dependent advection–diffusion problem by an implicit two-weight, three-point finite difference scheme. We extend the scheme proposed by Chadha and Madden (2011), to incorporate an optimal time step selection algorithm for the method. The optimal values of the weights involved in the scheme have been obtained by using the notion of an equivalent differential equation (Warming and Hyett, 1974), and, by enforcing certain constraints due to von Neumann stability and a discrete maximum principle, we obtain sharp bounds for optimal time-stepping in terms of mesh width and problem data. Furthermore, we formulate the time step selection as a root-finding problem, and use this to show the optimal time step exists and is unique. The resulting method, based on optimal values of weights and optimal time-stepping, is of fifth-order in space, and third-order in time. The results of extensive numerical experiments are presented for several classic test problems for various values of the advection and diffusion parameters. These show that our proposed method is superior (quantitatively and qualitatively) to other schemes that can be represented within the framework of two-weighted schemes. We also present comparisons with other high-order methods, and an application of the method to a problem with non-smooth data.

Shallow groundwater thermal sensitivity to climate change and land cover disturbances: derivation of analytical expressions and implications for stream temperature modeling

Abstract. Climate change is expected to increase stream temperatures and the projected warming may alter the spatial extent of habitat for cold-water fish and other aquatic taxa. Recent studies have proposed that stream thermal sensitivities, derived from short-term air temperature variations, can be employed to infer future stream warming due to long-term climate change. However, this approach does not consider the potential for streambed heat fluxes to increase due to gradual warming of the shallow subsurface. The temperature of shallow groundwater is particularly important for the thermal regimes of groundwater-dominated streams and rivers. Also, recent studies have investigated how land surface perturbations, such as wildfires or timber harvesting, can influence stream temperatures by changing stream surface heat fluxes, but these studies have typically not considered how these surface disturbances can also alter shallow groundwater temperatures and streambed heat fluxes. In this study, several analytical solutions to the one-dimensional unsteady advection–diffusion equation for subsurface heat transport are employed to estimate the timing and magnitude of groundwater temperature changes due to seasonal and long-term variability in land surface temperatures. Groundwater thermal sensitivity formulae are proposed that accommodate different surface warming scenarios. The thermal sensitivity formulae suggest that shallow groundwater will warm in response to climate change and other surface perturbations, but the timing and magnitude of the subsurface warming depends on the rate of surface warming, subsurface thermal properties, bulk aquifer depth, and groundwater velocity. The results also emphasize the difference between the thermal sensitivity of shallow groundwater to short-term (e.g., seasonal) and long-term (e.g., multi-decadal) land surface-temperature variability, and thus demonstrate the limitations of using short-term air and water temperature records to project future stream warming. Suggestions are provided for implementing these formulae in stream temperature models to accommodate groundwater warming.

A monolithic multi-time-step computational framework for first-order transient systems with disparate scales

Developing robust simulation tools for problems involving multiple mathematical scales has been a subject of great interest in computational mathematics and engineering. A desirable feature to have in a numerical formulation for multiscale transient problems is to be able to employ different time-steps (multi-time-step coupling), and different time integrators and different numerical formulations (mixed methods) in different regions of the computational domain. To this end, we present two new monolithic multi-time-step mixed coupling methods for first-order transient systems. We shall employ unsteady advection–diffusion–reaction equation with linear decay as the model problem, which offers several unique challenges in terms of non-self-adjoint spatial operator and rich features in the solutions. We shall employ the dual Schur domain decomposition technique to split the computational domain into an arbitrary number of subdomains. It will be shown that the governing equations of the decomposed problem, after spatial discretization, will be differential/algebraic equations. This is a crucial observation to obtain stable numerical results. Two different methods of enforcing compatibility along the subdomain interface will be used in the time discrete setting. A systematic theoretical analysis (which includes numerical stability, influence of perturbations, bounds on drift along the subdomain interface) will be performed. The first coupling method ensures that there is no drift along the subdomain interface, but does not facilitate explicit/implicit coupling. The second coupling method allows explicit/implicit coupling with controlled (but non-zero) drift in the solution along the subdomain interface. Several canonical problems will be solved to numerically verify the theoretical predictions, and to illustrate the overall performance of the proposed coupling methods. Finally, we shall illustrate the robustness of the proposed coupling methods using a multi-time-step transient simulation of a fast bimolecular advective–diffusive–reactive system.

Analytical solutions for one-dimensional advection–dispersion equation of the pollutant concentration

We present simple analytical solutions for the unsteady advection–dispersion equations describing the pollutant concentration C(x, t) in one dimension. The solutions are obtained by using Laplace transformation technique. In this study we divided the river into two regions x ≤ 0 and x≥0 and the origin at x = 0. The variation of C(x, t) with the time t from t = 0 up to t → ∞ (the steady state case) is taken into account in our study. The special case for which the dispersion coefficient D = 0 is studied in detail. The parameters controlling the pollutant concentration along the river are determined.

An a posteriori error estimator for an unsteady advection–diffusion–reaction problem

In this work we introduce an a posteriori error estimator, of the residual type, for the unsteady advection–diffusion–reaction problem. For the discretization in time we use an implicit Euler scheme and a continuous, piecewise linear triangular finite elements for the space together with a stabilized scheme. We prove that the approximation error is bounded, by above and below, by the error estimator. Using that, an adaptive algorithm is proposed, analyzed and tested numerically to prove the efficiency of our approach.

Stress development, relaxation, and memory in colloidal dispersions: Transient nonlinear microrheology

The motion of a single Brownian particle in a complex fluid can reveal material behavior both at and away from equilibrium. In active microrheology, a probe particle is driven by an external force through a complex medium and its motion studied in order to infer properties of the embedding material. Most work in microrheology has focused on steady behavior and established the relationship between the motion of the probe, the microstructure, and the effective microviscosity of the medium. Transient behavior in the near-equilibrium, linear-response regime has also been studied via its connection to low-amplitude oscillatory probe forcing and the complex modulus; at very weak forcing, the microstructural response that drives viscosity is indistinguishable from equilibrium fluctuations. But important information about the basic physical aspects of structural development and relaxation in a medium is captured by startup and cessation of the imposed deformation in the nonlinear regime, where the structure is driven far from equilibrium. Here, we study theoretically and by dynamic simulation the transient behavior of a colloidal dispersion undergoing nonlinear microrheological forcing. The strength with which the probe is forced, Fext, compared to thermal forces, kT/b, governs the dynamics and defines a Péclet number, Pe=Fext/(kT/b), where kT is the thermal energy and b is the colloidal bath particle size. For large Pe, a boundary layer (in which unsteady advection balances diffusion) forms at particle contact on the time scale of the flow, a/U, where a is the probe size and U its speed, whereas the wake forms over O(Pe) diffusive time steps. Similarly, relaxation following cessation occurs over several time scales corresponding to distinct physical processes. For very short times, the time scale for relaxation is set by a boundary layer of thickness δ∼(a+b)/Pe, and so τ∼δ2/Dr, where Dr is the relative diffusivity between the probe of size a and a bath particle. Nearly all stress relaxation occurs during this time. At longer times, the Brownian diffusion of the bath particles acts to close the wake on a time scale set by how long it takes a bath particle to diffuse laterally across it, τ∼(a+b)2/Dr. Although the majority of the microstructural relaxation occurs during this wake-healing process, it does so with little change in the stress. Also during relaxation, the probe travels backward in the suspension; this recovered strain is proportional to the free energy stored in the compressed particle configuration, an indicator that the stress is proportional to the free energy density stored entropically in the microstructure. Theoretical results are compared with Brownian dynamics simulation where it is found that the dilute theory captures the correct behavior even for concentrated suspensions. Two modes of forcing are studied: Constant force and constant velocity. Results are compared to analogous macrorheology results for suspensions undergoing simple shear flow.

Time-Centered Split Method for Implicit Discretization of Unsteady Advection Problems

A new method for implicit solution of unsteady nonlinear advection equations is presented. This time-centered split (TCS) method uses a nested application of the midpoint rule to computationally decouple advection terms in a temporally second-order accurate time-marching discretization. The method requires solution of only two sets of linear equations without an outer iteration, and is theoretically applicable to quadratically nonlinear coupled equations for any number of variables. The TCS algorithm is compared to other nonlinear solution methods (local linearization, Picard iteration, and Newton iteration) and applied to the Crank-Nicolson discretization of the one-dimensional Burgers’ equation. The temporal accuracy and practical stability of the method is confirmed using an unsteady flow test case with an analytical solution. The method is shown to require computational effort similar to local linearization, but does not require discrete computation of a functional Jacobian for solution.

Insights on a sign‐preserving numerical method for the advection–diffusion equation

AbstractIn this paper we explore theoretically and numerically the application of the advection transport algorithm introduced by Smolarkiewicz to the one‐dimensional unsteady advection–diffusion equation. The scheme consists of a sequence of upwind iterations, where the initial iteration is the first‐order accurate upwind scheme, while the subsequent iterations are designed to compensate for the truncation error of the preceding step. Two versions of the method are discussed. One, the classical version of the method, regards the second‐order terms of the truncation error and the other considers additionally the third‐order terms. Stability and convergence are discussed and the theoretical considerations are illustrated through numerical tests. The numerical tests will also indicate in which situations are advantageous to use the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd.

Numerical modelling of unsteady convective–diffusive heat transfer with a control volume hybrid method

Presented in this paper is a numerical methodology for the solution of the parabolic governing partial differential equation that describes unsteady advection–diffusion heat transfer. The formulation presented here is shown to be free from the numerical oscillation commonly associated with advection–diffusion heat transfer regardless of the value of the Peclet number. The formulation involves the absorption of the advection term in the unsteady heat equation into the capacitance term. This process is achieved with the use of a control volume methodology applied to each nodal element on a finite-volume mesh. This is shown to ensure that spurious energy losses and gains are avoided and provides for consistency between temperature and energy change. This approach provides unconditional stability and it is shown that good accuracy is achievable with relatively large time-steps. In order to highlight the features of the approach it is compared against those of benchmark numerical schemes. Detailed analysis is performed for the 1D semi-infinite moving solid problem for which an exact solution is available and for a realistic engineering heat transfer problem. Oscillation free results are achieved at good accuracy for a wide range of Peclet numbers and problems considered.

A discrete calculus analysis of the Keller Box scheme and a generalization of the method to arbitrary meshes

The Keller Box scheme is a face-based method for solving partial differential equations that has numerous attractive mathematical and physical properties. It is shown that these attractive properties collectively follow from the fact that the scheme discretizes partial derivatives exactly and only makes approximations in the algebraic constitutive relations appearing in the PDE. The exact discrete calculus associated with the Keller Box scheme is shown to be fundamentally different from all other mimetic (physics capturing) numerical methods. This suggests that a unique exact discrete calculus does not exist. It also suggests that existing analysis techniques based on concepts in algebraic topology (in particular – the discrete de Rham complex) are unnecessarily narrowly focused since they do not capture the Keller Box scheme. The discrete calculus analysis allows a generalization of the Keller Box scheme to non-simplectic meshes to be constructed. Analysis and tests of the method on the unsteady advection–diffusion equations are presented.