Discontinuous Flux Research Articles

A large-time-stepping scheme for balance equations

A well-balanced, large-time-stepping method for conservation laws with source terms is presented. The numerical method is based on a local reformulation of the balance law as a conservation law with a discontinuous flux function, and the approximate solution of this equation by a front tracking method. This yields an unconditionally stable method which is particularly well suited to calculate stationary states. The viability of this approach is demonstrated by several numerical examples.

Explicit Hopf–Lax type formulas for Hamilton–Jacobi equations and conservation laws with discontinuous coefficients

We deal with a Hamilton–Jacobi equation with a Hamiltonian that is discontinuous in the space variable. This is closely related to a conservation law with discontinuous flux. Recently, an entropy framework for single conservation laws with discontinuous flux has been developed which is based on the existence of infinitely many stable semigroups of entropy solutions based on an interface connection. In this paper, we characterize these infinite classes of solutions in terms of explicit Hopf–Lax type formulas which are obtained from the viscosity solutions of the corresponding Hamilton–Jacobi equation with discontinuous Hamiltonian. This also allows us to extend the framework of infinitely many classes of solutions to the Hamilton–Jacobi equation and obtain an alternative representation of the entropy solutions for the conservation law. We have considered the case where both the Hamiltonians are convex (concave). Furthermore, we also deal with the less explored case of sign changing coefficients in which one of the Hamiltonians is convex and the other concave. In fact in convex–concave case we cannot expect always an existence of a solution satisfying Rankine–Hugoniot condition across the interface. Therefore the concept of generalised Rankine–Hugoniot condition is introduced and prove existence and uniqueness.

Steady-state, control, and capacity calculations for flocculated suspensions in clarifier–thickeners

A spatially one-dimensional model for the behaviour of flocculated suspensions in clarifier–thickeners is studied. This model combines a theory of sedimentation–consolidation processes of flocculated suspensions, which leads to a strongly degenerate diffusion equation, with the discontinuous flux appearing in the recently analyzed clarifier–thickener (CT) setup. This setup includes both clarification and thickening zones of clarifier–thickener units, while the earlier ideal continuous thickener (ICT) concept explicitly models the thickening zone only. The construction of steady-state concentration profiles attainable in a continuously operated CT is described. This construction incorporates an entropy principle, which implies that only those steady states are admissible for which the solids concentration is increasing downwards. Numerical examples of steady-state profiles and their applications to comparisons between both modes of operation, to the control of sediment height through selection of the clarification/thickening split ratio of the feed flux, and for capacity calculations are presented. A numerical example illustrates the use of a numerical method for the full (time-dependent) model to compare several fill-up strategies.

A kinematic model of continuous separation and classification of polydisperse suspensions

Kinematic models for polydisperse suspensions are based on specifying the solid–fluid relative velocity for each solids species as a function of the local solids concentrations. One such model, the Masliyah–Lockett–Bassoon (MLB) model, is employed herein to simulate continuous separation and classification of polydisperse suspensions. To this end, the clarifier-thickener (CT) setup for the continuous separation of suspensions is extended to a generalized clarifier-thickener (GCT). Discharge streams (or products) are described by new singular sink terms. Combining the GCT setup with the MLB model yields a system of nonlinear conservation laws with a discontinuous flux and a new non-conservative transport term describing the sinks. A numerical algorithm for the solution of this equation is presented along with numerical examples. The model describes the GCT unit with all critical design parameters, and predicts the composition of the overflow, underflow and discharge streams and the spatio-temporal evolution of the solids species concentrations inside the unit.

Least-squares finite-element SN method for solving three-dimensional transport equation

A discrete ordinates finite-element method for solving three-dimensional first-order neutron transport equation is proposed using a least-squares variation. It avoids the singularity in void regions of the method derived from the second-order equation. Different from using the standard Galerkin variation applying to the first-order equation, the least-squares variation results in a symmetric matrix, which can be solved easily and effectively. The approach allows a continuous finite-element. To eliminate the discontinuity of the angular flux on the fixed flux boundary in the spherical harmonics method, the equation is discretized using the discrete ordinates method for angular dependency. A three-dimensional transport simulation code is developed and applied to some benchmark problems with unstructured geometry. The numerical results demonstrate the accuracy and feasibility of the method.

Convergence of Godunov type methods for a conservation law with a spatially varying discontinuous flux function

We deal with single conservation laws with a spatially varying and possibly discontinuous coefficient. This equation includes as a special case single conservation laws with conservative and possibly singular source terms. We extend the framework of optimal entropy solutions for these classes of equations based on a two-step approach. In the first step, an interface connection vector is used to define infinite classes of entropy solutions. We show that each of these classes of solutions is stable in L1. This allows for the possibility of choosing one of these classes of solutions based on the physics of the problem. In the second step, we define optimal entropy solutions based on the solution of a certain optimization problem at the discontinuities of the coefficient. This method leads to optimal entropy solutions that are consistent with physically observed solutions in two-phase flows in heterogeneous porous media. Another central aim of this paper is to develop suitable numerical schemes for these equations. We develop and analyze a set of Godunov type finite volume methods that are based on exact solutions of the corresponding Riemann problem. Numerical experiments are shown comparing the performance of these schemes on a set of test problems.

A least-squares finite-element Sn method for solving first-order neutron transport equation

A discrete ordinates finite-element method for solving the two-dimensional first-order neutron transport equation is derived using the least-squares variation. It avoids the singularity in void regions of the method derived from the second-order equation which contains the inversion of the cross-section. Different from using the standard Galerkin variation to the first-order equation, the least-squares variation results in a symmetric matrix, which can be solved easily and effectively. To eliminate the discontinuity of the angular flux on the vacuum boundary in the spherical harmonics method, the angle variable is discretized by the discrete ordinates method. A two-dimensional transport simulation code is developed and applied to some benchmark problems with unstructured geometry. The numerical results verified the validity of this method.

A well posed conservation law with a variable unilateral constraint

This paper considers the Cauchy problem for a conservation law with a variable unilateral constraint, its motivation being, for instance, the modeling of a toll gate along a highway. This problem is solved by means of nonclassical shocks and its well posedness is proved. Then, the solutions so obtained are shown to coincide with the limits of the classical solutions to suitable conservation laws with discontinuous flux function that approximate the constrained problem.

Conservation laws with discontinuous flux

We consider a hyperbolic conservation law with discontinuous flux. Such a partial differential equation arises in different applications, in particular we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak entropic solutions.

Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes

We consider a scalar conservation law with a discontinuous flux function. The fluxes are non-convex, have multiple points of extrema and can have arbitrary intersections. We propose an entropy formulation based on interface connections and associated jump conditions at the interface. We show that the entropy solutions with respect to each choice of interface connection exist and form a contractive semi-group in $L^1$. Existence is shown by proving convergence of a Godunov type scheme by a suitable modification of the singular mapping approach. This extends the results of [3] to the general case of non-convex flux geometries.

Evidence for the influence of agriculture on weather and climate through the transformation and management of vegetation: Illustrated by examples from the Canadian Prairies

This paper lists selected observational and modeling studies which provide evidence that agriculture, through the transformation and management of vegetation, has had, and continues to have, an impact upon the weather and climate on the local, regional and global scales. The influence of agriculture on weather and climate, through the alteration of the physiological properties of the land cover, is illustrated by examples from the cropped grassland of the Canadian Prairies. The physiological and physical properties of the vegetation, along with the land cover's impact upon the level of available soil moisture, affect the weather and climate by influencing the transfer of heat, moisture and momentum from the land surface to the overlying air. The principle physiological properties are leaf area, stomatal resistance, and rooting depth; the main physical properties are albedo and surface roughness. By land clearing, cultivation and the grazing of domesticated animals, man has transformed and now manages the vegetation over vast areas of the globe. Agriculture influences the availability of energy and water vapour mass for moist deep convection on the local and regional scales. By creating latent heat flux discontinuities, it may induce mesoscale circulations that initiate moist deep convection. Agriculture, by affecting the level of stored soil moisture, moisture that is available to the vegetation during a later period, may influence the level of convective activity within a region during a subsequent season. Thus agriculture, through the physiological and physical properties of the land cover, has had, and continues to have, an impact upon near surface weather elements and, more significantly, upon the regional hydrologic cycle. Spatially coherent and persistent patterns of thunderstorms play a role in the export of heat and moisture from lower to higher latitudes—this may effect the general circulation. Thus agriculture, by influencing the occurrence, location and intensity of moist deep convection, particularly in the tropics, may also influence global weather and climate.

Interaction of elementary waves of scalar conservation laws with discontinuous flux function

In this paper, the Riemann solutions for scalar conservation laws with discontinuous flux function were constructed. The interactions of elementary waves of the conservation laws were concerned, and the numerical simulations were given.

A domain decomposition method for conservation laws with discontinuous flux function

A domain decomposition method for scalar conservation laws with discontinuous flux function is presented. The spatial domain is decomposed in subdomains at the location of the discontinuity of the flux function. The associated conservation law for each subdomain having then a continuous flux function is solved using a nonoscillatory relaxation scheme. Special coupling conditions at the subdomain interfaces are developed based on the direction of traveling waves. This method uses similar techniques recently used to treat junctions in network problems. The numerical performance of the method is illustrated on typical one-dimensional test examples like traffic models or two-phase flows.

On an extended clarifier-thickener model with singular source and sink terms

A one-dimensional model of clarifier-thickener units in engineering applications can be expressed as a conservation law with a flux that is discontinuous with respect to the spatial variable. This model also includes a singular feed source. In this paper, the clarifier-thickener model studied in a previous paper (Numer. Math.97 (2004) 25–65) is extended by a singular sink through which material is extracted from the unit. A difficulty is that in contrast to the singular source, the sink term cannot be incorporated into the flux function; rather, the sink is represented by a new non-conservative transport term. To focus on the new analytical difficulties arising due to this non-conservative term, a reduced problem is formulated, which contains the new sink term of the extended clarifier-thickener model, but not the source term and flux discontinuities. The paper is concerned with numerical methods for both models (extended and reduced) and with the well-posedness analysis for the reduced problem. For the reduced problem, a definition of entropy solutions, based on Kružkov-type entropy functions and fluxes, is provided. Jump conditions are derived and uniqueness of the entropy solution is shown. Existence of an entropy solution is shown by proving convergence of a monotone difference scheme. Two variants of the numerical scheme are introduced. Numerical examples illustrate that all three variants converge to the entropy solution, but introduce different amounts of numerical diffusion.

A mathematical model of clarifier‐thickener units

AbstractWe review a recent analysis of a clarifier‐thickener model for flocculated suspensions. The resulting governing PDE is a strongly degenerate convection‐diffusion equation with discontinuous flux. A new numerical example illustrates how control actions speed up thickener operation significantly. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Stable numerical methods for conservation laws with discontinuous flux function

We develop numerical methods for solving nonlinear equations of conservation laws with flux function that depends on discontinuous coefficients. Using a relaxation approximation, the nonlinear equation is transformed to a semilinear diagonalizable problem with linear characteristic variables. Eulerian and Lagrangian methods are used for the advection stage while an implicit–explicit scheme solves the relaxation stage. The main advantages of this approach are neither Riemann problem solvers nor linear iterations are required during the solution process. Moreover, the characteristic-based relaxation method is unconditionally stable such that no CFL conditions are imposed on the selection of time steps. Numerical results are shown for models on traffic flows and two-phase flows.

Solutions to a Scalar Discontinuous Conservation Law in a Limit Case of Phase Transitions

For a particular discontinuous flux function that can be associated to the limit case of a phase transition, we introduce an appropriate notion of entropy weak solution to the Cauchy problem for the corresponding conservation law. Then, for a class of initial data, that includes the Riemann data, we prove, by the vanishing viscosity method and with a suitable regularisation of the flux function, the existence of an entropy weak solution. This result can be easily extended to more general flux functions.

Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies

We prove uniqueness of solutions to scalar conservation laws with space discontinuous fluxes. To do so, we introduce a partial adaptation of Kružkov's entropies which naturally takes into account the space dependency of the flux. The advantage of this approach is that the proof turns out to be a simple variant of the original method of Kružkov. In particular, we do not need traces, interface conditions, bounded variation assumptions (neither on the solution nor on the flux), or convex fluxes. However, we use a special ‘local uniform invertibility’ structure of the flux, which applies to cases where different interface conditions are known to yield different solutions

On the approximation of the solutions of the Riemann problem for a discontinuous conservation law

For a class of discontinuous flux functions introduced in [3] (cf. also [4]), we prove, for the Riemann problem, an extension of the existence result proved in [2] for a Lipschitz continuous flux function. In the last section, and based in the previous results, we apply the Lax-Friedrichs approximation method and the limiters technique (cf.[6]) to compute the quoted solution in a numerical example. For related results see [5].

Convergence of Upwind Finite Difference Schemes for a Scalar Conservation Law with Indefinite Discontinuities in the Flux Function

We consider the scalar conservation law with flux function discontinuous in the space variable, i.e., \begin{eqnarray} \label{eq1} u_t+(H(x)f(u)+(1-H(x))g(u))_{x} &=& 0 \quad \mbox{in } \R \times \R_{+}, \nonumber \\ u(0, x) &=& u_{0}(x) \quad \mbox{in } \R, \label{0.1} \end{eqnarray} where H is the Heaviside function and f and g are smooth with the assumptions that either f is convex and g is concave or f is concave and g is convex. The existence of a weak solution of (\ref{eq1}) is proved by showing that upwind finite difference schemes of Godunov and Enquist--Osher type converge to a weak solution. Uniqueness follows from a Kruzkhov-type entropy condition. We also provide explicit solutions to the Riemann problem for (\ref{eq1}). At the level of numerics, we give easy-to-implement numerical schemes of Godunov and Enquist--Osher type. The central feature of this paper is the modification of the singular mapping technique (the main analytical tool for these types of equations) which allows us to show that the numerical schemes converge. Equations of type (\ref{eq1}) with the above hypothesis on the flux may occur when considering the following scalar conservation law with discontinuous flux: \begin{equation} \label{eq2} \begin{array}{r@{\;}l} u_t + (k (x) f (u))_x &= 0, \\ u (0, x) &= u_0 (x), \end{array} \end{equation} with f convex and k of indefinite sign.