Space-dependent Flux Research Articles

Stability of a numerical scheme for methane transport in hydrate zone under equilibrium and non-equilibrium conditions

In this paper we carry out numerical analysis for a family of simplified gas transport models with hydrate formation and dissociation in subsurface, in equilibrium and non-equilibrium conditions. These models are adequate for simulation of hydrate phase change at basin and at shorter time scales, but the analysis does not account directly for the related effects of evolving hydraulic properties. To our knowledge this is the first analysis of such a model. It is carried out for the transport steps while keeping the pressure solution fixed. We frame the transport model as conservation law with a non-smooth space-dependent flux function; the kinetic model approximates this equilibrium. We prove weak stability of the upwind scheme applied to the regularized conservation law. We illustrate the model, confirm convergence with numerical simulations, and illustrate its use for some relevant equilibrium and non-equilibrium scenarios.

Microscopic and macroscopic traffic flow models including random accidents

We introduce microscopic and macroscopic stochastic traffic models including traffic accidents. The microscopic model is based on a Follow-the-Leader approach whereas the macroscopic model is described by a scalar conservation law with space dependent flux function. Accidents are introduced as interruptions of a deterministic evolution and are directly linked to the traffic situation. Based on a Lax-Friedrichs discretization convergence of the microscopic model to the macroscopic model is shown. Numerical simulations are presented to compare the above models and show their convergence behaviour.

Modeling random traffic accidents by conservation laws.

We introduce a stochastic traffic flow model to describe random traffic accidents on a single road. The model is a piecewise deterministic process incorporating traffic accidents and is based on a scalar conservation law with space-dependent flux function. Using a Lax-Friedrichs discretization, we show that the total variation is bounded in finite time and provide a theoretical framework to embed the stochastic process. Additionally, a solution algorithm is introduced to also investigate the model numerically.

Convergence of the follow-the-leader scheme for scalar conservation laws with space dependent flux

This paper deals with the derivation of entropy solutions to Cauchy problems for a class of scalar conservation laws with space-density depending fluxes from systems of deterministic particles of follow-the-leader type. We consider fluxes which are product of a function of the density \begin{document}$ v(\rho) $\end{document} and a function of the space variable \begin{document}$ \phi(x) $\end{document} . We cover four distinct cases in terms of the sign of \begin{document}$ \phi $\end{document} , including cases in which the latter is not constant. The convergence result relies on a local maximum principle and on a uniform \begin{document}$ BV $\end{document} estimate for the approximating density.

A finite volume method for scalar conservation laws with stochastic time–space dependent flux functions

We propose a new finite volume method for scalar conservation laws with stochastic time–space dependent flux functions. The stochastic effects appear in the flux function and can be interpreted as a random manner to localize the discontinuity in the time–space dependent flux function. The location of the interface between the fluxes can be obtained by solving a system of stochastic differential equations for the velocity fluctuation and displacement variable. In this paper we develop a modified Rusanov method for the reconstruction of numerical fluxes in the finite volume discretization. To solve the system of stochastic differential equations for the interface we apply a second-order Runge–Kutta scheme. Numerical results are presented for stochastic problems in traffic flow and two-phase flow applications. It is found that the proposed finite volume method offers a robust and accurate approach for solving scalar conservation laws with stochastic time–space dependent flux functions.

Numerical Methods for Balance Laws with Space Dependent Flux: Application to Radiotherapy Dose Calculation

AbstractThe present work is concerned with the derivation of numerical methods to approximate the radiation dose in external beam radiotherapy. To address this issue, we consider a moment approximation of radiative transfer, closed by an entropy minimization principle. The model under consideration is governed by a system of hyperbolic equations in conservation form supplemented by source terms. The main difficulty coming from the numerical approximation of this system is an explicit space dependence in the flux function. Indeed, this dependence will be seen to be stiff and specific numerical strategies must be derived in order to obtain the needed accuracy. A first approach is developed considering the 1D case, where a judicious change of variables allows to eliminate the space dependence in the flux function. This is not possible in multi-D. We therefore reinterpret the 1D scheme as a scheme on two meshes, and generalize this to 2D by alternating transformations between separate meshes. We call this procedure projection method. Several numerical experiments, coming from medical physics, illustrate the potential applicability of the developed method.

Hyperbolic conservation laws with space-dependent fluxes: II. General study of numerical fluxes

Following the previous paper, this one continues to study numerical approximations to the space-dependent flux functions in hyperbolic conservation laws. The investigation is based on the wave propagation behavior, Riemann problem, steady flows, hyperbolic properties, cell entropy inequalities, along with such well known numerical fluxes as the Godunov, Local Lax–Friedrichs and Engquist–Osher. All these give rise to correct description for the consistency and monotonicity of numerical fluxes, which ensure properly confined numerical solutions. Numerical examples show that the accordingly designed fluxes resolve discontinuities and smooth solutions very precisely.

Hyperbolic conservation laws with space-dependent flux: I. Characteristics theory and Riemann problem

In the paper, a kind of one-dimensional scalar hyperbolic conservation laws with flux functions dependent on space variable is discussed and analyzed. A better understanding about the behavior of wave propagation of the kind problems is presented. Especially, some sufficient and necessary conditions that ensure the unique physically relevant solution to the Riemann problem are proposed. Because the numerical flux obtained from the Riemann's solver is theoretically correct and exact to the problem, it must also be of high resolution in its nature. For comparison, some convincing numerical examples from traffic flow problems are given at the end of the paper.

Space-Dependent Resonance Self-Shielding

A new method is developed to determine space-dependent, self-shielded cross sections for resonance nuclides with no overlapping resonances, contained in an arbitrarily shaped absorber body within some general lattice configuration. The theoretical basis for the method is discussed, and analytical expressions are presented for the space-dependent flux spectrum in the vicinity of an isolated resonance and for the space-dependent variation in the shielded resonance integral and multigroup cross section. The shielded cross-section expressions contain space-dependent, “weighted escape probabilities” that correspond to the weighted average of the energy-dependent escape probability over each energy group. The method is implemented in an assembly lattice physics code, and results are compared to those obtained with a highly accurate transport theory calculation that uses pointwise cross-section data. The method gives good agreement for the radial variation in the self-shielded cross section through a boiling water reactor fuel pellet.

Calculation of the void reactivity of CANDU lattices using the scale code system

The reactivity effect of coolant voiding in CANDU-type fuel lattices has been calculated with different methods using the scale code system. The known positive void reactivity coefficient of the original lattice was correctly obtained. A modified fuel bundle containing dysprosium and slightly enriched uranium to eliminate the positive reactivity effect was also calculated. Owing to the increased heterogeneity of this modified fuel the one-dimensional cylindrical calculation with XSDRN proved to be inadequate. Code options allowing bundle geometry were successfully used for the calculation of the strongly space dependent flux and spectrum changes which determine the void reactivity.

Dynamic model for cross-flow microfiltration of microbial suspensions in porous tubes

A theoretical development is presented which describes the time- and space-dependent flux during cross-flow filtration of a microbial suspension under laminar flow conditions. The treatment is based on the numerical solution of the set of non-stationary momentum and material balance equations. A time moving boundary condition at the porous wall describes the dynamics of cake build-up and flux decline. In addition, the concept of shear-induced diffusion has been applied for modelling the shear-induced transport of microbial cells.

Interference between resonances in a slab lattice

In calculating resonance absorption for one resonance, the effect of neighbouring resonances is usually neglected. It is, therefore, customary to assume that the slowing-down flux takes its asymptotic value above the resonant energy and becomes independent of both lethargy and space. However, when the resonances are not widely separated, a resonance may lie in the region of energy oscillation due to other high-energy resonances. In this region the flux may depend on both lethargy and space. In the present paper a method has been developed for estimating this error. In this calculation the higher energy resonance has been replaced by a delta function sink, and the age-diffusion equation with modified source term has been solved to obtain the lethargy and space-dependent flux. The resonance integrals have then been calculated in the presence and in the absence of the higher energy resonance in an effort to estimate the importance of this effect. Various calculations of the effect of resonance interference have ignored spatial effects, but these may be important, particularly for tightly packed lattices.