1D Transport Equation Research Articles

On a 1D transport equation with nonlocal velocity and supercritical dissipation

We study a 1D transport equation with nonlocal velocity. First, we prove eventual regularization of the viscous regularization when dissipation is in the supercritical range with non-negative initial data. Next, we will prove global regularity for solutions when dissipation is slightly supercritical. Both results utilize a nonlocal maximum principle.

Estimation of open channels hydraulic parameters with the stochastic particle collision algorithm

In the present work, the 1D flow and transport equations for open channels are numerically solved and coupled to a recently developed global search optimization, the particle collision algorithm (PCA), to estimate two essential parameters present in flow and transport equations, respectively, the bed roughness and the dispersion coefficient. The PCA is inspired in the scattering and absorption phenomena of a given incident nuclear particle by a target nucleus. In this method, if the particle in a given location of the design space reaches a low value of the objective function, it is absorbed, otherwise, it is scattered. This allows the search space to be widely explored, in such a way that the most promising regions are searched through successive scattering and absorption events. Based on real data measured in the Albear channel, Cuba, the bed roughness and longitudinal dispersion coefficient were successfully estimated from two numerical experiments dealing, respectively, with flow and transport equations. The results obtained were supported by the high correlations achieved between simulations and observations, demonstrating the feasibility of the approach here considered.

NUMERICAL SIMULATION OF POLLUTANT TRANSPORT FROM FISH FARMING IN RIVER

This paper presented a 3D model for substance transport in river and its application for simulation of pollutant transport in Mekong river due to floating cages-raising. 3D flow-field was solved by logarithmic distributing 2D flow-field of averaged height. Pollutant transport is calculated by solving its full 3D transport equation. The 2D continuum and momentum equations were solved by finite difference method with ADI scheme of Ponce-Yabusaki. The 3D transport equation was solved by finite volume method with ADI scheme of Douglas-Gunn in sigma transformed co-ordinate. The model was tested over analytical solution. Some preliminary results of simulation for pollutant transport of Myhoahung floating cages area (Angiang province) are also presented.

Pseudo-two-dimensional Poisson equation for the modeling of field-effect transistors

A modified Poisson equation able to take into account the influence of gates and a ?-doping in FETs and HEMTs is proposed. This equation can be solved self-consistently together with 1D transport equations along inhomogeneous transistor channels like those of the hydrodynamic or drift-diffusion approximations or with a Monte Carlo simulator used to describe carrier transport in n+nn+ structures.

A new 1D approximation for the solution of 2D radiative transfer problems

Abstract The simplified M-1D algorithm (M stands for modified) for the calculation of horizontal distribution of reflected brightness coefficient in 2D regions with large homogeneous pixels is presented. This algorithm is based upon modified 3⁎N−1 1D-transport equations (where N is the number of large pixels) instead of one 2D-transport equation, usually used in such problems. The method does not rely on empiric assumptions on both optical properties of atmosphere or diffuse radiation intensity. Numerical results demonstrating the accuracy of the presented algorithm for simulating brightening and shadowing effects in a vicinity of the jump of optical properties given. The accuracy of the M-1D approximation strongly depends on the geometry and illumination conditions. However, it remains below 15% and can reach 1% for all cases studied and is strongly higher than the accuracy of usually employed independent pixel approximation. Time reduction via replacing 2D problem via modified 1D problem is about 17 times for all cases considered in this paper.

On 1D diffusion problems with a gradient-dependent diffusion coefficient

0021-9991/$ see front matter 2008 Elsevier Inc doi:10.1016/j.jcp.2008.06.032 * Corresponding author. Tel.: +1 609 243 2635; fa E-mail address: jardin@pppl.gov (S.C. Jardin). In solving the 1D (flux surface averaged) transport equations for the temperatures, magnetic fields, and densities in the ‘‘evolving equilibrium” description of a tokamak [1], one increasingly encounters highly nonlinear thermal conductivity and diffusivity functions, such as GLF23 [2], that have a strong and non-analytic dependence on the temperature gradients. These arise from a subsidiary microstability based calculation in which the growth rates and hence transport coefficients are sensitive functions of these gradients [3]. When these nonlinear functions are interfaced with an existing transport framework that uses a standard implicit time advancement algorithm such as Crank-Nicolson or backward Euler [4], large non-physical oscillations can develop and, as a result, non-convergent solutions can occur. Here we describe a relatively simple modification to these implicit algorithms that cures this difficulty. To illustrate the method, we start with a simple diffusion equation in cylindrical polar coordinates:

Analyses of Solute Transport Using Streamline Simulation and Semi-analytical Solutions

In this article, we present a two-dimentional (2D) solute transport model using streamline simulation and semi-analytical solutions of one-dimensional (1D) transport equation. We validated the accuracy of the developed model by comparing three simulation results of contaminant transport using an analytical, a semi-analytical, and an upwind finite-difference solution of 1D transport equation. We also show how to apply the developed model to simulate solute transport with complex boundary conditions, such as the first-order degraded source condition and the Cauchy boundary condition. For an advection-dominant flow, we can clearly see the differences of plume spreads between the streamline simulation models with the semi-analytical solution and the numerical solution. From the comparison of computing time, the streamline simulation using the semi-analytical solution is 10–200 times faster than the streamline simulation using the numerical solution. If we would use a semi-analytical solution instead of a numerical solution in case no analytical solution is available, we could obtain faster and more accurate simulation results free from numerical dispersion.

Blow-up of solutions for a 1D transport equation with nonlocal velocity and supercritical dissipation

We study a 1D transport equation with nonlocal velocity and supercritical dissipation. We show that for a certain class of initial data the solution blows up in finite time.

Multirate-Transfer Dual-Porosity Modeling of Gravity Drainage and Imbibition

SummaryWe develop a physically motivated approach to modeling displacement processes in fractured reservoirs. To find matrix/ fracture transfer functions in a dual-porosity model, we use analytical expressions for the average recovery as a function of time for gas gravity drainage and countercurrent imbibition. For capillary-controlled displacement, the recovery tends to its ultimate value with an approximately exponential decay (Barenblatt et al. 1990). When gravity dominates, the approach to ultimate recovery is slower and varies as a power law with time (Hagoort 1980). We apply transfer functions based on these expressions for core-scale recovery in field-scale simulation.To account for heterogeneity in wettability, matrix permeability, and fracture geometry within a single gridblock, we propose a multirate model (Ponting 2004). We allow the matrix to be composed of a series of separate domains in communication with different fracture sets with different rate constants in the transfer function.We use this methodology to simulate recovery in a Chinese oil field to assess the efficiency of different injection processes. We use a streamline-based formulation that elegantly allows the transfer between fracture and matrix to be accommodated as source terms in the 1D transport equations along streamlines that capture the flow in the fractures (Di Donato et al. 2003; Di Donato and Blunt 2004; Huang et al. 2004).This approach contrasts with the current Darcy-like formulation for fracture/matrix transfer based on a shape factor (Gilman and Kazemi 1983) that may not give the correct average behavior (Di Donato et al. 2003; Di Donato and Blunt 2004; Huang et al. 2004). Furthermore, we show that recovery is exceptionally sensitive to parameters that describe the physics of the displacement process, highlighting the need to make careful core-scale measurements of recovery.

Formation of singularities for a transport equation with nonlocal velocity

We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time.

Formation of singularities for a transport equation with nonlocal velocity

We study a 1D transport equation with nonlocal velocity and show the formation of singularities in finite time for a generic family of initial data. By adding a diffusion term the finite time singularity is prevented and the solutions exist globally in time.

Three high-order splitting schemes for 3D transport equation

Two high-order splitting schemes based on the idea of the operators splitting method are given. The three-dimensional advection-diffusion equation was split into several one-dimensional equations that were solved by these two schemes, only three computational grid points were needed in each direction but the accuracy reaches the spatial fourth-order. The third scheme proposed is based on the classical ADI scheme and the accuracy of the advection term of it can reach the spatial fourth-order. Finally, two typical numerical experiments show that the solutions of these three schemes compare well with that given by the analytical solution when the Peclet number is not bigger than 5.

Analysis of the effects of geometry on the fluorescence radiation field in the frame of transport theory

AbstractIn x‐ray fluorescence spectroscopy, a photon beam is focused on the sample to stimulate the emission of characteristic radiation. Even if a qualitative interpretation of the measurements is simple, a quantitative analysis is not straightforward because the primary photons are produced deep in the target and the properties of the radiation that reaches the detector are modified significantly by the interactions undergone before leaving the specimen. Understanding how the emission spectra are influenced by interactions with matter is a central problem in fluorescence analysis. In this work, by using the 3D transport equation, we found that not only the composition of the specimen but also the geometry of the system plays an important role in determining the properties of the radiation field, denoting by geometry the shape of the target, the direction of the incoming beam and the observation angle. Copyright © 2004 John Wiley & Sons, Ltd.

Overland Water Flow and Solute Transport: Model Development and Field-Data Analysis

The application of plant nutrients with irrigation water is an efficient and cost-effective method for fertilizer application to enhance crop production and reduce or eliminate potential environmental problems related to conventional application methods. In this study, a combined overland water flow and solute transport model for analysis and management of surface fertigation/chemigation is presented. Water flow is predicted with the well-known Saint-Venant's equations using a control volume of moving cells, while solute transport is modeled with the advection-dispersion equation. The 1D transport equation was solved using a Crank-Nicholson finite- difference scheme. Four, large-scale, field experiments were conducted on blocked-end and free draining furrows to calibrate and verify the proposed model. The results showed that application of solute during the entire irrigation event, or during the second half of the irrigation for blocked end conditions with appropriate inflow rates, produced higher solute uniformity than application of solute during the first half of the irrigation event. Measured fertilizer distribution uniformity of the low quarter ranged from 21 to 76% while fertilizer distribution uniformity of the low half values varied between 62 to 87%. The model was subsequently applied to the experimental data; results showed good agreement with all field data. Water balance errors for the different experiments varied from 0.004 to 1.8%, whereas fertilizer mass balance errors ranged from 1.2 to 3.6%. A sensitivity analysis was also performed to assess the effects of longitudinal dispersivity parameter on overland solute concentrations. A value of 10 cm for dispersivity provided a reasonable fit to the experimental data.

Nonlinear and Linear α-Weighted Methods for Particle Transport Problems

A parametrized family of iterative methods for the planar-geometry transport equation is proposed. This family is a generalization of previously proposed nonlinear flux methods. The new methods are derived by integrating the 1D transport equation over −1≤μ≤0 and 0≤μ≤1 with weight |μ|α, α≥0. Both nonlinear and linear methods are developed. The convergence properties of the proposed methods are studied theoretically by means of a Fourier stability analysis. The optimum value of α that provides the best convergence rate is derived. We also show that the convergence rates of nonlinear and linear methods are almost the same. Numerical results are presented to confirm these theoretical predictions.

Local Galerkin's schemes qualitative analysis for 1D transport equation solving

Using the local projection method on the test function space associated to the elementary cell (Δχ,Δμ), the transport equation is reduced to a matrix equation. The method used here is called Green Matrix Method (GMM). For the S N approximation, the GMM formalism represents a generalization of the difference scheme without using the diamond relation. With GMM, the spatial integration presents truncation errors due to the consideration of a finite number of terms in the development of the angular flux in Legendre-Fourier series. For the S N approximation, on the base of the analytical expressions of the Green Matrix elements (GME), we qualitatively evaluate the GMM performances. Unlike the diamond difference scheme, the GMM algorithm presents a new type of truncation error for thin spatial meshes. This error is due to the approximate representation in rational fraction of the GME. Using symbolical methods, we also treat the two-cyclic iteration scheme for the S 4 and DP 1 approximations. The two-cyclic algorithms developed both an improvement of precision and a substantial increase of the convergence rate comparatively to the classical iteration schema on the scattering source. Numerical tests present the dependence of the results precision on the spatial approximation order. The GMM is superior in point of precision comparatively to the diamond difference scheme.

Pencil beam transport in a random medium using the Fermi transport equation

The effect of a random transport cross section on the transverse spreading of a pencil beam of charged particles is analysed using the Fermi approximation to the 3D transport equation, in conjunction with a Gaussian model of cross section fluctuations. Stochasticity is seen to enhance the spreading of the beam but ensemble averaged quantities such as the transverse spatial moments and the scalar flux appear not to be sensitive to the details of the underlying statistical model. Furthermore, only asymptotic solutions for the scalar flux are possible because of divergent behaviour that can be traced to an inconsistency between the assumption of forward peakedness of the angular flux. which underlies the Fermi approximation, and the spreading induced by the randomness of the material properties.

Transient 1D transport equation simulated by a mixed Green element formulation

New discrete element equations or coefficients are derived for the transient 1D diffusion–advection or transport equation based on the Green element replication of the differential equation using linear elements. The Green element method (GEM), which solves the singular boundary integral theory (a Fredholm integral equation of the second kind) on a typical element, gives rise to a banded global coefficient matrix which is amenable to efficient matrix solvers. It is herein derived for the transient 1D transport equation with uniform and non-uniform ambient flow conditions and in which first-order decay of the containment is allowed to take place. Because the GEM implements the singular boundary integral theory within each element at a time, the integrations are carried out in exact fashion, thereby making the application of the boundary integral theory more utilitarian. This system of discrete equations, presented herein for the first time, using linear interpolating functions in the spatial dimensions shows promising stable characteristics for advection-dominant transport. Three numerical examples are used to demonstrate the capabilities of the method. The second-order-correct Crank–Nicolson scheme and the modified fully implicit scheme with a difference weighting value of two give superior solutions in all simulated examples. © 1997 John Wiley & Sons, Ltd.

Transient 1D transport equation simulated by a mixed Green element formulation

Transient 1D transport equation simulated by a mixed Green element formulation

Analytic structure of two 1D-transport equations with nonlocal fluxes

We replace the flux term in Burger's equation by two simple alternates that contain contributions depending globally on the solution. In one case, the term is in the form of a hyperbolic equation where the characteristic speed is nonlocal, and in the other the term is in conservation form. In both cases, the nonanalytic is due to the presence of the Hilbert transform. The equations have a loose analogy to the motion of vortex sheets. In particular, they both form singularities in finite time in the absence of viscous effects. Our motivation then is to study the influence of viscosity. In one case, viscosity does not prevent singularity formation. In the other, we can prove solutions exist for all time, and determine the likely weak solution as viscosity vanishes. An interesting aspect of our work is that singularity formation can be viewed as the motion of singularities in the complex physical plane that reach the real axis in finite time. In one case, the singularity is a pole and causes the solution to blow up when it reaches the real axis. In the other, numerical solutions and an asymptotic analysis suggest that the weak solution contains a square root singularity that reaches the real axis in finite time, and then propagates along it. We hope our results will spur further interest in the role of singularities in the complex spatial plane in solutions to transport equations.