Discontinuous Flux Research Articles

Applicability of angular flux discontinuity factor preserving region-wise leakage for integro-differential transport equation

In the current core analysis, spatial homogenization is utilized to reduce the computational time. The discontinuity factor (DF) is one of the effective correction factors to reduce spatial homogenization error. The DF in diffusion equation is widely used; on the other hand the DF in transport equation has not been put to practical use although several efforts have been carried out. In this paper, the angular flux discontinuity factor (AFDF) as the DF for the integro-differential transport equation (e.g., the discrete-ordinate method, the method of characteristics) is theoretically described and its applicability is discussed. The AFDF is used to preserve the region-wise neutron leakage at each spatial mesh and defined as a ratio of heterogeneous and homogeneous angular fluxes at the homogenized region surface. In a homogeneous calculation with the AFDF, the angular flux is discontinuous at the region surface. In this paper the applicability of the AFDF to fuel pin cell homogenization is verified for one-dimensional slab geometry. As a result of this verification, it is confirmed that the AFDF has the capability to reduce the spatial homogenization error of fuel pin cell homogenization.

Instability of Riemann solutions to a scalar conservation law with discontinuous flux

This paper is devoted to studying the structure stability and instability of Riemann solutions to a scalar conser- vation law with a flux function involving discontinuous coefficients under certain small perturbations on the discontinuous coefficient k and the unknown function u. More precisely, the initial data are taken as three piecewise constant states, and the middle region is regarded as the perturbed region with small distance. We can see that the Riemann solutions may be unstable with respect to the small perturbations and the sufficient and necessary condition is also given. The study is based on the so-called double Riemann problem, and wave interactions are considered in detail by employing the method of characteristics.

A robust and contact resolving Riemann solver on unstructured mesh, Part II, ALE method

This paper investigates solution behaviors under the strong shock interaction for moving mesh schemes based on the one-dimensional Godunov and HLLC Riemann solvers. When the grid motion velocity is close to Lagrangian one, these Godunov methods, which updates the flow parameters directly on the moving mesh without using interpolation, may suffer from numerical shock instability. In order to cure such instability, a new cell centered arbitrary Lagrangian Eulerian (ALE) algorithm is constructed for inviscid, compressible gas flows. The main feature of the algorithm is to introduce a nodal contact velocity and ensure the compatibility between edge fluxes and the nodal flow intrinsically. We establish a new two-dimensional Riemann solver based on the HLLC method (denoted by the ALE HLLC-2D). The solver relaxes the condition that the contact pressures must be the same in the traditional HLLC solver and constructs discontinuous fluxes along each sampling direction of the similarity solution. The two-dimensional contact velocity of the grid node is determined via enforcing conservation of mass, momentum and total energy. The resulting ALE scheme has a node instead of grid edge conservation properties. Numerical tests are presented to demonstrate the robustness and accuracy of this new solver. Due to the multi-dimensional information introduced and consistency between the fluxes and nodal contact velocity, the developed ALE algorithm performs well on both quadrilateral and triangular grids and reduces numerical shock instability phenomena.

Second order scheme for scalar conservation laws with discontinuous flux

Burger, Karlsen, Torres and Towers in [9] proposed a flux TVD (FTVD) second order scheme with Engquist–Osher flux, by using a new nonlocal limiter algorithm for scalar conservation laws with discontinuous flux modeling clarifier thickener units. In this work we show that their idea can be used to construct FTVD second order scheme for general fluxes like Godunov, Engquist–Osher, Lax–Friedrich,… satisfying (A, B)-interface entropy condition for a scalar conservation law with discontinuous flux with proper modification at the interface. Also corresponding convergence analysis is shown. We show further from numerical experiments that solutions obtained from these schemes are comparable with the second order schemes obtained from the minimod limiter.

A robust and contact resolving Riemann solver on unstructured mesh, Part I, Euler method

This article presents a new cell-centered numerical method for compressible flows on arbitrary unstructured meshes. A multi-dimensional Riemann solver based on the HLLC method (denoted by HLLC-2D solver) is established. The work is an extension from the cell-centered Lagrangian scheme of Maire et al. [27] to the Eulerian framework. Similarly to the work in [27], a two-dimensional contact velocity defined on a grid node is introduced, and the motivation is to keep an edge flux consistency with the node velocity connected to the edge intrinsically. The main new feature of the algorithm is to relax the condition that the contact pressures must be same in the traditional HLLC solver. The discontinuous fluxes are constructed across each wave sampling direction rather than only along the contact wave direction. The two-dimensional contact velocity of the grid node is determined via enforcing conservation of mass, momentum and total energy, and thus the new method satisfies these conservation properties at nodes rather than on grid edges. Other good properties of the HLLC-2d solver, such as the positivity and the contact preserving, are described, and the two-dimensional high-order extension is constructed employing MUSCL type reconstruction procedure. Numerical results based on both quadrilateral and triangular grids are presented to demonstrate the robustness and the accuracy of this new solver, which shows it has better performance than the existing HLLC method.

An easy-to-use algorithm for simulating traffic flow on networks: Numerical experiments

In this paper we propose a Godunov-based discretization of a hyperbolic system of conservation laws with discontinuous flux, modeling vehicular flow on a network. Each equation describes the density evolution of vehicles having a common path along the network. We show that the algorithm selects automatically an admissible solution at junctions, hence <em>ad hoc</em> external procedures (e.g., maximization of the flux via a linear programming method) usually employed in classical approaches are no needed. Since users have not to deal explicitly with vehicle dynamics at junction, the numerical code can be implemented in minutes. We perform a detailed numerical comparison with a Godunov-based scheme coming from the classical theory of traffic flow on networks which maximizes the flux at junctions.

An easy-to-use algorithm for simulating traffic flow on networks: Theoretical study

In this paper we study a model for traffic flow on networks based on a hyperbolic system of conservation laws with discontinuous flux. Each equation describes the density evolution of vehicles having a common path along the network. In this formulation the junctions disappear since each path is considered as a single uninterrupted road. <br>&nbsp; &nbsp We consider a Godunov-based approximation scheme for the system which is very easy to implement. Besides basic properties like the conservation of cars and positive bounded solutions, the scheme exhibits other nice properties, being able to select automatically a solution at network's nodes without requiring external procedures (e.g., maximization of the flux via a linear programming method). Moreover, the scheme can be interpreted as a discretization of the traffic models with buffer, although no buffer is introduced here. <br>&nbsp; &nbsp Finally, we show how the scheme can be recast in the framework of the classical theory of traffic flow on networks, where a conservation law has to be solved on each arc of the network. This is achieved by solving the Riemann problem for a modified equation, and showing that its solution corresponds to the one computed by the numerical scheme.

A dissipation-rate reserving DG method for wave catching-up phenomena in a nonlinearly elastic composite bar

A tensile wave can be destructive, as it can generate microcracks and cause the interface detachments in a composite structure. Recently Huang, Dai, Chen and Kong [14] proposed to use material nonlinearity to generate wave catching-up phenomena such that a tensile wave can catch the first transmitted compressive wave, leading to the reduction of the former. They gave the existence and asymptotic solutions in short time. In this paper, we develop a dissipation-rate reserving discontinuous Galerkin (DG) method to solve numerically the system of balance laws with discontinuous flux and inflow boundary condition for waves in a two-material nonlinearly elastic composite bar. Physically, the aim is to examine the reduction ratio of the tensile wave due to wave catching-up phenomena in long time. The stability analysis for the DG method is also presented. And, a contribution is that the numerical fluxes are chosen to satisfy the local dissipation rate across a shock. Numerical results are compared with the available asymptotic solutions in short time and good agreements are found. In particular, the numerical solutions give a high resolution for various interactions between the tensile shock, compressive shock and rarefaction wave. The results confirm that when the stress-strain relation of the second nonlinearly elastic material is convex, indeed the tensile wave can catch the previously transmitted compressive wave. The simulation results by the DG method show that the catching-up phenomena can reduce the magnitude of the tensile wave by more than 400%. We also examine how material parameters influence the reduction ratio. As a verification, the solutions by a weighted essentially non-oscillatory (WENO) finite volume method are also presented, which are in agreement with those by the DG method.

DIFFUSION PIECEWISE HOMOGENIZATION VIA FLUX DISCONTINUITY RATIOS

We analyze piecewise homogenization with flux-weighted cross sections and preservation of averaged currents at the boundary of the homogenized domain. Introduction of a set of flux discontinuity ratios (FDR) that preserve reference interface currents leads to preservation of averaged region reaction rates and fluxes. We consider the class of numerical discretizations with one degree of freedom per volume and per surface and prove that when the homogenization and computing meshes are equal there is a unique solution for the FDRs which exactly preserve interface currents. For diffusion submeshing we introduce a Jacobian-Free Newton-Krylov method and for all cases considered obtain an ‘exact’ numerical solution (eight digits for the interface currents). The homogenization is completed by extending the familiar full assembly homogenization via flux discontinuity factors to the sides of regions laying on the boundary of the piecewise homogenized domain. Finally, for the familiar nodal discretization we numerically find that the FDRs obtained with no submesh (nearly at no cost) can be effectively used for whole-core diffusion calculations with submesh. This is not the case, however, for cell-centered finite differences.

Numerical modelling of three-phase immiscible flow in heterogeneous porous media with gravitational effects

This paper presents a new numerical formulation for the simulation of immiscible and incompressible three-phase water–gas–oil flows in heterogeneous porous media. We take into account the gravitational effects, both variable permeability and porosity of porous medium, and explicit spatially varying capillary pressure, in the diffusive fluxes, and explicit spatially varying flux functions, in the hyperbolic operator. The new formulation is a sequential time marching fractional-step procedure based in a splitting technique to decouple the equations with mixed discretization techniques for each of the subproblems: convection, diffusion, and pressure–velocity. The system of nonlinear hyperbolic equations that models the convective transport of the fluid phases is approximated by a modified central scheme to take into account the explicit spatially discontinuous flux functions and the effects of spatially variable porosity. This scheme is coupled with a locally conservative mixed finite element formulation for solving parabolic and elliptic problems, associated respectively with the diffusive transport of fluid phases and the pressure–velocity problem. The time discretization of the parabolic problem is performed by means of an implicit backward Euler procedure. The hybrid-mixed formulation reported here is designed to handle discontinuous capillary pressures. The new method is used to numerically investigate the question of existence, and structurally stable, of three-phase flow solutions for immiscible displacements in heterogeneous porous media with gravitational effects. Our findings appear to be consistent with theoretical and experimental results available in the literature.

Sensitivity Analysis of Optimal Operation of an Activated Sludge Process Model for Economic Controlled Variable Selection

This paper describes a systematic sensitivity analysis of optimal operation conducted on an activated sludge process model based on the test-bed benchmark simulation model no. 1 (BSM1) and the activated sludge model no. 1 (ASM1). The objective is to search for a control structure that leads to optimal economic operation, while promptly rejecting disturbances at lower layers in the control hierarchy avoiding thus violation of the more important regulation constraints on effluent discharge. We start by optimizing a steady-state nonlinear model of the process. Here, a new steady-state secondary settler mathematical model is developed based on the theory of partial differential equations applied to the conservation law with discontinuous fluxes. The resulting active constraints must be chosen as economic controlled variables. These are the effluent ammonia from the bioreaction section and the final effluent total suspended solids at their respective upper limits, in addition to the internal recycle flow rate ...

A numerical method for two-phase flow in fractured porous media with non-matching grids

We propose a novel computational method for the efficient simulation of two-phase flow in fractured porous media. Instead of refining the grid to capture the flow along the faults or fractures, we represent the latter as immersed interfaces, using a reduced model for the flow and suitable coupling conditions. We allow for non matching grids between the porous matrix and the fractures to increase the flexibility of the method in realistic cases. We employ the extended finite element method for the Darcy problem and a finite volume method that is able to handle cut cells and matrix-fracture interactions for the saturation equation. Moreover, we address through numerical experiments the problem of the choice of a suitable numerical flux in the case of a discontinuous flux function at the interface between the fracture and the porous matrix. A wrong approximate solution of the Riemann problem can yield unphysical solutions even in simple cases.

Riemann solver for a kinematic wave traffic model with discontinuous flux

We investigate a model for traffic flow based on the Lighthill–Whitham–Richards model that consists of a hyperbolic conservation law with a discontinuous, piecewise-linear flux. A mollifier is used to smooth out the discontinuity in the flux function over a small distance ϵ≪1 and then the analytical solution to the corresponding Riemann problem is derived in the limit as ϵ→0. For certain initial data, the Riemann problem can give rise to zero waves that propagate with infinite speed but have zero strength. We propose a Godunov-type numerical scheme that avoids the otherwise severely restrictive CFL constraint that would arise from waves with infinite speed by exchanging information between local Riemann problems and thereby incorporating the effects of zero waves directly into the Riemann solver. Numerical simulations are provided to illustrate the behavior of zero waves and their impact on the solution. The effectiveness of our approach is demonstrated through a careful convergence study and comparisons to computations using a third-order WENO scheme.

Well-Posedness for a One-Dimensional Fluid-Particle Interaction Model

The fluid-particle interaction model introduced by the three last authors in [J. Differential Equations, 245 (2008), pp. 3503-3544] is the object of our study. This system consists of the Burgers equation with a singular source term (term that models the interaction via a drag force with a moving point particle) and of an ODE for the particle path. The notion of entropy solution for the singular Burgers equation is inspired by the theory of conservation laws with discontinuous flux developed by the first author, Kenneth Hvistendahl Karlsen and Nils Henrik Risebro in [Arch. Ration. Mech. Anal., 201 (2011), pp. 26-86]. In this paper, we prove well-posedness and justify an approximation strategy for the particle-in-Burgers system in the case of initial data of bounded variation. Existence result for L∞ data is also given.

On the singular limit of a two‐phase flow equation with heterogeneities and dynamic capillary pressure

AbstractWe consider conservation laws with spatially discontinuous flux that are perturbed by diffusion and dispersion terms. These equations arise in a theory of two‐phase flow in porous media that includes rate‐dependent (dynamic) capillary pressure and spatial heterogeneities. We investigate the singular limit as the diffusion and dispersion parameters tend to zero, showing strong convergence towards a weak solution of the limit conservation law.

MULTI-DIMENSIONAL SCALAR CONSERVATION LAWS WITH FLUXES DISCONTINUOUS IN THE UNKNOWN AND THE SPATIAL VARIABLE

The paper deals with a scalar conservation law in an arbitrary dimension d with a discontinuous flux. The flux is supposed to be a discontinuous function in the spatial variable x and in an unknown function u. Under some additional hypothesis on the structure of possible discontinuities, we formulate an appropriate notion of entropy solution and establish its existence and uniqueness. The framework for proving the existence and uniqueness of entropy weak solutions is provided by the studies on entropy measure-valued solutions and may be viewed as a corollary of the uniqueness theorem for entropy measure-valued solutions.

A REDUCED-ORDER ODE–PDE MODEL FOR THE ACTIVATED SLUDGE PROCESS IN WASTEWATER TREATMENT: CLASSIFICATION AND STABILITY OF STEADY STATES

Most wastewater treatment plants contain an activated sludge process, which consists of a biological reactor and a sedimentation tank. The purpose is to reduce the incoming organic material and dissolved nutrients (the substrate). This is done in the biological reactor where micro-organisms (the biomass) decompose the substrate. The biomass is then separated from the water in the sedimentation tank under continuous in- and outflows. One of the outflows is recirculated to the reactor. The governing mathematical model describes the concentration of substrate and biomass as functions of time for the biological reactor, and as functions of time and depth for the sedimentation tank. This gives rise to a system of two ODEs for the reactor coupled with two spatially one-dimensional PDEs for the sedimentation tank. The main mathematical difficulty lies in the nonlinear PDE modeling the continuous sedimentation of the biomass. Previous analyses of models of the activated sludge process have included excessively simplifying assumptions on the sedimentation process. In this paper, results for nonlinear hyperbolic conservation laws with spatially discontinuous flux function are used to obtain a classification of the steady states for the coupled system. Their stability to disturbances are investigated and some phenomena are demonstrated by a numerical simulation.

Vanishing capillarity solutions of Buckley–Leverett equation with gravity in two-rocks’ medium

For the hyperbolic conservation laws with discontinuous-flux function, there may exist several consistent notions of entropy solutions; the difference between them lies in the choice of the coupling across the flux discontinuity interface. In the context of Buckley–Leverett equations, each notion of solution is uniquely determined by the choice of a “connection,” which is the unique stationary solution that takes the form of an under-compressive shock at the interface. To select the appropriate connection, following Kaasschieter (Comput Geosci 3(1):23–48, 1999), we use the parabolic model with small parameter that accounts for capillary effects. While it has been recognized in Cancès (Networks Het Media 5(3):635–647, 2010) that the “optimal” connection and the “barrier” connection may appear at the vanishing capillarity limit, we show that the intermediate connections can be relevant and the right notion of solution depends on the physical configuration. In particular, we stress the fact that the “optimal” entropy condition is not always the appropriate one (contrarily to the erroneous interpretation of Kaasschieter’s results which is sometimes encountered in the literature). We give a simple procedure that permits to determine the appropriate connection in terms of the flux profiles and capillary pressure profiles present in the model. This information is used to construct a finite volume numerical method for the Buckley–Leverett equation with interface coupling that retains information from the vanishing capillarity model. We support the theoretical result with numerical examples that illustrate the high efficiency of the algorithm.

Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes

Flow of two phases in a heterogeneous porous medium is modeled by a scalar conservation law with a discontinuous coefficient. As solutions of conservation laws with discontinuous coefficients depend explicitly on the underlying small scale effects, we consider a model where the relevant small scale effect is dynamic capillary pressure. We prove that the limit of vanishing dynamic capillary pressure exists and is a weak solution of the corresponding scalar conservation law with discontinuous coefficient. A robust numerical scheme for approximating the resulting limit solutions is introduced. Numerical experiments show that the scheme is able to approximate interesting solution features such as propagating non-classical shock waves as well as discontinuous standing waves efficiently.

A Numerical Method for Transport Equations with Discontinuous Flux Functions: Application to Mathematical Modeling of Cell Dynamics

In this work, we propose a numerical method to handle discontinuous fluxes arising in transport-like equations. More precisely, we study hyperbolic PDEs with flux transmission conditions at interfaces between subdomains where coefficients are discontinuous. A dedicated finite volume scheme with a limited high order enhancement is adapted to treat the discontinuities arising at interfaces. The validation of the method is done on one- and two-dimensional toy problems for which exact solutions are available, allowing us to do a thorough convergence study. We then apply the method to a biological model focusing on complex cell dynamics that initially motivated this study and illustrates the full potentialities of the scheme.