Discontinuous Flux Research Articles

Multi-scale homogenization of moving interface problems with flux jumps: application to solidification

In this paper, a multi-scale analysis scheme for solidification based on two-scale computational homogenization is discussed. Solidification problems involve evolution of surfaces coupled with flux jump boundary conditions across interfaces. We provide consistent macro-micro transition and averaging rules based on Hill’s macro- homogeneity condition. The overall macro-scale behavior is analyzed with solidification at the micro-scale modeled using an enthalpy formulation. The method is versatile in the sense that two different models can be employed at the macro- and micro-scales. The micro-scale model can incorporate all the physics associated with solidification including moving interfaces and flux discontinuities, while the macro-scale model needs to only model thermal conduction using continuous (homogenized) fields. The convergence behavior of the tightly coupled macro-micro finite element scheme with respect to decreasing element size is analyzed by comparing with a known analytical solution of the Stefan problem.

Zero Diffusion‐Dispersion‐Smoothing Limits for a Scalar Conservation Law with Discontinuous Flux Function

We consider multidimensional conservation laws with discontinuous flux, which are regularized with vanishing diffusion and dispersion terms and with smoothing of the flux discontinuities. We use the approach of H‐measures to investigate the zero diffusion‐dispersion‐smoothing limit.

On the compactness for two dimensional scalar conservation law with discontinuous flux

We prove that a family of solutions to a Cauchy problem for a two dimensional scalar conservation law with a discontinuous smoothed flux and the vanishing viscosity is strongly Lloc– precompact under a new genuine nonlinearity condition, weaker than in previous works on the subject.

An Engquist–Osher-Type Scheme for Conservation Laws with Discontinuous Flux Adapted to Flux Connections

We consider scalar conservation laws with the spatially varying flux $H(x)f(u)+(1-H(x))g(u)$, where $H(x)$ is the Heaviside function and f and g are smooth nonlinear functions. Adimurthi, Mishra, and Veerappa Gowda [J. Hyperbolic Differ. Equ., 2 (2005), pp. 783–837] pointed out that such a conservation law admits many $L^1$ contraction semigroups, one for each so-called connection $(A,B)$. Here we define entropy solutions of type $(A,B)$ involving Kružkov-type entropy inequalities that can be adapted to any fixed connection $(A,B)$. It is proved that these entropy inequalities imply the $L^1$ contraction property for $L^\infty$ solutions, in contrast to the “piecewise smooth” setting of Adimurthi, Mishra, and Veerappa Gowda. For a fixed connection, these entropy inequalities include a single adapted entropy of the type used by Audusse and Perthame [Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), pp. 253–265]. We prove convergence of a new difference scheme that approximates entropy solutions of type $(A,B)$ for any connection $(A,B)$ if a few parameters are varied. The scheme relies on a modification of the standard Engquist–Osher flux, is simple as no $2\times2$ Riemann solver is involved, and is designed such that the steady-state solution connecting A to B is preserved. In contrast to most analyses of similar problems, our convergence proof is not based on the singular mapping or compensated compactness methods, but on standard spatial variation estimates away from the flux discontinuity. Some numerical examples are presented.

A Molecular Dynamics Study on the Effects of Nanostructural Clearances on Thermal Resistance at a Liquid Water–Solid Interface

The thermal resistance at a liquid–solid interface was calculated by means of the heat flux and temperature discontinuity obtained through molecular dynamics simulations. The thermal resistance at a liquid–solid interface reached a minimum value when the interfacial structural clearances were changed with nanometer precision. The thermal resistance calculated by the rotational temperature of water molecules was larger than that calculated by the translational temperature because of the restriction on the dynamic motions of water molecules between the nanostructures. The interfacial thermal resistance reduction depended on the clearances and was related to the residence time variation of water molecules at the interface.

Multigroup pin power reconstruction with two-dimensional source expansion and corner flux discontinuity

A pin power reconstruction method that is readily applicable to multigroup problems with superior accuracy is presented for applications involving rectangular fuel assemblies. It employs a two-dimensional (2D), fourth order Legendre expansion of the source distribution that naturally leads to a group-decoupled, 2D semi-analytic solution of the neutron diffusion equation. The four surface average currents and four corner fluxes are used as the boundary conditions to uniquely specify the homogenous solution. The corner fluxes and source expansion coefficients are iteratively determined using the condition of corner point balance and the orthogonal property of the Lengedre functions. Corner discontinuity is incorporated in the calculation of the corner fluxes which turns out to be very effective in the cases of enrichment zoning. The accuracy of the proposed method is assessed by performing the two-step core calculations for the L336C5, C5G7MOX, and MOX core transient benchmark problems and then by comparing with the direct whole-core transport solutions. The results indicate that the proposed method is as accurate as the fully analytic method and works well irrespective the number of groups. However, it is also noted that somewhat larger errors are inevitable at the peripheral assemblies near the reflector in which the error associated with a prioi generation of the homogenized cross-sections and form functions is not trivial.

Hyperbolic conservation laws with discontinuous fluxes and hydrodynamic limit for particle systems

We study the following class of scalar hyperbolic conservation laws with discontinuous fluxes: (0.1) ∂ t ρ + ∂ x F ( x , ρ ) = 0 . The main feature of such a conservation law is the discontinuity of the flux function in the space variable x. Kruzkov's approach for the L 1 -contraction does not apply since it requires the Lipschitz continuity of the flux function in x; an additional jump wave may occur in the solution besides the classical waves; and entropy solutions even for the Riemann problem are not unique under the classical entropy conditions. On the other hand, it is known that, in statistical mechanics, some microscopic interacting particle systems with discontinuous speed-parameter λ ( x ) in the hydrodynamic limit formally lead to scalar hyperbolic conservation laws with discontinuous fluxes of the form (0.2) ∂ t ρ + ∂ x ( λ ( x ) h ( ρ ) ) = 0 . The natural question arises which entropy solution the hydrodynamic limit selects, thereby leading to a suitable, physical relevant notion of entropy solutions of this class of conservation laws. This paper is a first step and provides an answer to this question for a family of discontinuous flux functions. In particular, we identify the entropy condition for (0.1) and proceed to show the well-posedness by combining our existence result with a uniqueness result of Audusse and Perthame (2005) for the family of flux functions; we establish a compactness framework for the hydrodynamic limit of large particle systems and the convergence of other approximate solutions to (0.1), which is based on the notion and reduction of measure-valued entropy solutions; and we finally establish the hydrodynamic limit for a ZRP with discontinuous speed-parameter governed by an L ∞ entropy solution to (0.2).

Well-balanced schemes for conservation laws with source terms based on a local discontinuous flux formulation

We propose and analyze a finite volume scheme of the Godunov type for conservation laws with source terms that preserve discrete steady states. The scheme works in the resonant regime as well as for problems with discontinuous flux. Moreover, an additional modification of the scheme is not required to resolve transients, and solutions of non-linear algebraic equations are not involved. Our well-balanced scheme is based on modifying the flux function locally to account for the source term and to use a numerical scheme especially designed for conservation laws with discontinuous flux. Due to the difficulty of obtaining BV estimates, we use the compensated compactness method to prove that the scheme converges to the unique entropy solution as the discretization parameter tends to zero. We include numerical experiments in order to show the features of the scheme and how it compares with a wellbalanced scheme from the literature.

A discontinuous Galerkin method for the shallow water equations in spherical triangular coordinates

A global model of the atmosphere is presented governed by the shallow water equations and discretized by a Runge–Kutta discontinuous Galerkin method on an unstructured triangular grid. The shallow water equations on the sphere, a two-dimensional surface in R 3 , are locally represented in terms of spherical triangular coordinates, the appropriate local coordinate mappings on triangles. On every triangular grid element, this leads to a two-dimensional representation of tangential momentum and therefore only two discrete momentum equations. The discontinuous Galerkin method consists of an integral formulation which requires both area (elements) and line (element faces) integrals. Here, we use a Rusanov numerical flux to resolve the discontinuous fluxes at the element faces. A strong stability-preserving third-order Runge–Kutta method is applied for the time discretization. The polynomial space of order k on each curved triangle of the grid is characterized by a Lagrange basis and requires high-order quadature rules for the integration over elements and element faces. For the presented method no mass matrix inversion is necessary, except in a preprocessing step. The validation of the atmospheric model has been done considering standard tests from Williamson et al. [D.L. Williamson, J.B. Drake, J.J. Hack, R. Jakob, P.N. Swarztrauber, A standard test set for numerical approximations to the shallow water equations in spherical geometry, J. Comput. Phys. 102 (1992) 211–224], unsteady analytical solutions of the nonlinear shallow water equations and a barotropic instability caused by an initial perturbation of a jet stream. A convergence rate of O ( Δ x k + 1 ) was observed in the model experiments. Furthermore, a numerical experiment is presented, for which the third-order time-integration method limits the model error. Thus, the time step Δ t is restricted by both the CFL-condition and accuracy demands. Conservation of mass was shown up to machine precision and energy conservation converges for both increasing grid resolution and increasing polynomial order k.

Convergence of implicit Finite Volume methods for scalar conservation laws with discontinuous flux function

This paper deals with the problem of numerical approximation in the Cauchy-Dirichlet problem for a scalar conservation law with a flux function having finitely many discontinuities. The well-posedness of this problem was proved by Carrillo [J. Evol. Eq. 3 (2003) 687–705]. Classical numerical methods do not allow us to compute a numerical solution (due to the lack of regularity of the flux). Therefore, we propose an implicit Finite Volume method based on an equivalent formulation of the initial problem. We show the well-posedness of the scheme and the convergence of the numerical solution to the entropy solution of the continuous problem. Numerical simulations are presented in the framework of Riemann problems related to discontinuous transport equation, discontinuous Burgers equation, discontinuous LWR equation and discontinuous non-autonomous Buckley-Leverett equation (lubrication theory).

Two-region Milne problem with anisotropic scattering

We consider the two-region Milne problem, defined as the steady-state monoenergetic linear transport problem for two adjoining homogeneous source-free half-spaces, with particle source coming from infinity in the positive half-space. The integral version of the transport equation is solved using trial functions based on Case’s eigen modes and exponential integral function. The solution of the Milne problem is formulated in terms of characteristic parameters such as the extrapolation length, the fractional scalar flux discontinuity and the fractional current discontinuity. Numerical results for the analytically evaluated parameters are then presented. Some of our numerical results are compared with the available published results.

Narrative, splintered temporalities and the unconscious in 20th century music

Narrative structures the human experience of time, but does it also organise our musical experience? Behind this question lies another one, which concerns the narrative process itself: does it belong solely to the time of consciousness or does it manifest itself through other forms of temporal organisation in the unconscious mind? Psychologists have identified a structure of the experience of time that precedes narrative itself, which can be called “proto-narrative form” and which organises the coherence and unfolding of narrative, as it does perhaps the unfolding of musical form. It may be characterised by its linearity and a strong directionality, implying a clearly perceptible and temporally oriented line of dramatic tension. However, during the 20th century, directionality and linearity have progressively given way to a-directional or poly-directional fragmented forms, implying discontinuities of the temporal flux and the superimposition of multiple lines of dramatic tension that have neither the same progressions nor the same endings. What sense can we give to these new splintered forms of time? We attempt to answer this question from a psychoanalytic perspective.

Conservation laws with discontinuous flux: a short introduction

Conservation laws with discontinuous flux have attracted recent attention both due to their numerous applications and the intriguing theoretical challenges raised by their well-posedness and numerical analysis. This introductory note states the basic problem considered in the eight contributions of this Special Issue. Three different types of applications are surveyed where these equations appear, motivated by spatially heterogeneous physical models, adjoint problems for parameter identification, and numerical methods for systems of conservation laws, respectively. Basic problems arising in the analysis of these equations are discussed, and the contributions of the Special Issue are presented.

Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model

The classical Lighthill-Whitham-Richards (LWR) kinematic traffic model is extended to a unidirectional road on which the maximum density $a(x)$ represents road inhomogeneities, such as variable numbers of lanes, and is allowed to vary discontinuously. The car density $\phi = \phi(x,t)$ is then determined by the following initial value problem for a scalar conservation law with a spatially discontinuous flux: $\phi_t + (\phi v(\phi/{a(x)})_x = 0, \quad \phi(x,0)=\phi_0(x),\quad x \in \mathbb{R},\quad t\in (0,T),$ (*) where $v(z)$ is the velocity function. We adapt to (*) a new notion of entropy solutions (Burger, Karlsen, and Towers [Submitted, 2007]), which involves a Kružkov-type entropy inequality based on a specific flux connection $(A,B)$, and which we interpret in terms of traffic flow. This concept is consistent with both the driver's ride impulse and the desire of drivers to speed up. We prove that entropy solutions of type $(A,B)$ are unique. This solution concept also leads to simple, transparent, and unified convergence proofs for numerical schemes. Indeed, we adjust to (*) new variants of the Engquist-Osher (EO) scheme (Burger, Karlsen, and Towers [Submitted, 2007]), and of the Hilliges-Weidlich (HW) scheme analyzed by the authors [ J. Engrg. Math. , to appear]. It is proven that the EO and HW schemes and a related Godunov scheme converge to the unique entropy solution of type $(A,B)$ of (*). For the Godunov version, this is the first rigorous convergence and well-posedness result, since no unnecessarily restrictive regularity assumptions are imposed on the solution. Numerical experiments for first-order schemes and formally second-order MUSCL/Runge-Kutta versions are presented.

Numerical schemes for kinematic flows with discontinuous flux

AbstractSpatially one‐dimensional kinematic flows arise in a series of applications including traffic flow and sedimentation. They lead to nonlinear systems of conservation law whose flux has an explicit “concentration times velocity” structure. A new family of simple numerical schemes which are adapted to this structure, and which handle fluxes that are discontinuous with respect to the space variable, is presented and in part analyzed. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Multiresolution schemes for an extended clarifier‐thickener model

AbstractA fully adaptive finite volume multiresolution scheme for one‐dimensional strongly degenerate parabolic equations with discontinuous flux modelling an extended clarifier‐thickener, is presented. The numerical scheme is based on a finite volume discretization using the approximation of Engquist‐Osher for the flux and explicit time stepping. Cell averages multiresolution scheme speeds up CPU time and memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A space–time discontinuous Galerkin finite element method for fully coupled linear thermo-elasto-dynamic problems with strain and heat flux discontinuities

A discontinuous Galerkin finite element method for the solution of linear thermo-elasto-dynamic problems is proposed and its unconditional stability without any restrictions on the grid structure is proven. Applications to space–time problems with discontinuous data are demonstrated.

A family of numerical schemes for kinematic flows with discontinuous flux

Multiphase flows of suspensions and emulsions are frequently approximated by spatially one-dimensional kinematic models, in which the velocity of each species of the disperse phase is an explicitly given function of the vector of concentrations of all species. The continuity equations for all species then form a system of conservation laws which describes spatial segregation and the creation of areas of different composition. This class of models also includes multi-class traffic flow, where vehicles belong to different classes according to their preferential velocities. Recently, these models were extended to fluxes that depend discontinuously on the spatial coordinate, which appear in clarifier–thickener models, in duct flows with abruptly varying cross-sectional area, and in traffic flow with variable road surface conditions. This paper presents a new family of numerical schemes for such kinematic flows with a discontinuous flux. It is shown how a very simple scheme for the scalar case, which is adapted to the “concentration times velocity” structure of the flux, can be extended to kinematic models with phase velocities that change sign, flows with two or more species (the system case), and discontinuous fluxes. In addition, a MUSCL-type upgrade in combination with a Runge–Kutta-type time discretization can be devised to attain second-order accuracy. It is proved that two particular schemes within the family, which apply to systems of conservation laws, preserve an invariant region of admissible concentration vectors, provided that all velocities have the same sign. Moreover, for the relevant case of a multiplicative flux discontinuity and a constant maximum density, it is proved that one scalar version converges to a BV t entropy solution of the model. In the latter case, the compactness proof involves a novel uniform but local estimate of the spatial total variation of the approximate solutions. Numerical examples illustrate the performance of all variants within the new family of schemes, including applications to problems of sedimentation, traffic flow, and the settling of oil-in-water emulsions.

The Analytic Coarse-Mesh Finite Difference Method for Multigroup and Multidimensional Diffusion Calculations

In this work we develop and demonstrate the analytic coarse-mesh finite difference (ACMFD) method for multigroup—with any number of groups—and multidimensional diffusion calculations of eigenvalue and external source problems. The first step in this method is to reduce the coupled system of the G multigroup diffusion equations, inside any homogenized region (or node) of any size, to the G independent modal equations in the real or complex eigenspace of the G × G multigroup matrix. The mathematical and numerical analysis of this step is discussed for several reactor media and number of groups.As a second step, we discuss the analytical solutions in the general (complex) modal eigenspace for one-dimensional plane geometry, deriving the generalized Chao’s relation among the surface fluxes and the net currents, at a given interface, and the node-average fluxes, essential in the ACMFD method. We also introduce here the treatment of heterogeneous nodes, through modal interface flux discontinuity factors, and show the analytical and numerical application to core-reflector problems, for a single infinite reflector and for reflectors with two layers of different materials.Then, we address the general multidimensional case, with rectangular X-Y-Z geometry considered, showing the equivalency of the methods of transverse integration and incomplete expansion of the multidimensional fluxes, in the real or complex modal eigenspace of the multigroup matrix. A nonlinear iteration scheme is implemented to solve the multigroup multidimensional nodal problem, which has shown a fast and robust convergence in proof-of-principle numerical applications to realistic pressurized water reactor cores, with heterogeneous fuel assemblies and reflectors.

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the Engquist-Osher approximation for the flux and explicit time-stepping. An adaptive multiresolution scheme with cell averages is then used to speed up CPU time and meet memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for the sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier-thickener model illustrate the efficiency of this method.