Antidiffusive Lagrangian‐remap schemes for models of polydisperse sedimentation

One‐dimensional models of gravity‐driven sedimentation of polydisperse suspensions with particles that belong to N size classes give rise to systems of N strongly coupled, nonlinear first‐order conservation laws for the local solids volume fractions. As the eigenvalues and eigenvectors of the flux Jacobian have no closed algebraic form, characteristic‐wise numerical schemes for these models become involved. Alternative simple schemes for this model directly utilize the velocity functions and are based on splitting the system of conservation laws into two different first‐order quasi‐linear systems, which are solved successively for each time iteration, namely, the Lagrangian and remap steps (so‐called Lagrangian‐remap [LR] schemes). This approach was advanced in (Bürger, Chalons, and Villada, SIAM J Sci Comput 35 (2013), B1341–B1368) for a multiclass Lighthill–Whitham‐Richards traffic model with nonnegative velocities. By incorporating recent antidiffusive techniques for transport equations a new version of these Lagrangian‐antidiffusive remap (L‐AR) schemes for the polydisperse sedimentation model is constructed. These L‐AR schemes are supported by a partial analysis for N = 1. They are total variation diminishing under a suitable CFL condition and therefore converge to a weak solution. Numerical examples illustrate that these schemes, including a more accurate version based on MUSCL extrapolation, are competitive in accuracy and efficiency with several existing schemes. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1109–1136, 2016     

Polynomial viscosity methods for multispecies kinematic flow models

Multispecies kinematic flow models are defined by systems of strongly coupled, nonlinear first‐order conservation laws. They arise in various applications including sedimentation of polydisperse suspensions and multiclass vehicular traffic. Their numerical approximation is a challenge since the eigenvalues and eigenvectors of the corresponding flux Jacobian matrix have no closed algebraic form. It is demonstrated that a recently introduced class of fast first‐order finite volume solvers, called polynomial viscosity matrix (PVM) methods [M. J. Castro Díaz and E. Fernández‐Nieto, SIAM J Sci Comput 34 (2012), A2173–A2196], can be adapted to multispecies kinematic flows. PVM methods have the advantage that they only need some information about the eigenvalues of the flux Jacobian, and no spectral decomposition of a Roe matrix is needed. In fact, the so‐called interlacing property (of eigenvalues with known velocity functions), which holds for several important multispecies kinematic flow models, provides sufficient information for the implementation of PVM methods. Several variants of PVM methods (differing in polynomial degree and the underlying quadrature formula to approximate the Roe matrix) are compared by numerical experiments. It turns out that PVM methods are competitive in accuracy and efficiency with several existing methods, including the Harten, Lax, and van Leer method and a spectral weighted essentially non‐oscillatory scheme that is based on the same interlacing property. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1265–1288, 2016     

On linearly implicit IMEX Runge-Kutta methods for degenerate convection-diffusion problems modeling polydisperse sedimentation

Implicit-explicit (IMEX) Runge-Kutta (RK) methods are suitable for the solution of nonlinear, possibly strongly degenerate, convection-diffusion problems, since the stability restrictions, coming from the explicitly treated convective part, are much less severe than those that would be deduced from an explicit treatment of the diffusive term. A particularly efficient variant of these schemes, so-called linearly implicit IMEX-RK schemes, arise from discretizing the diffusion terms in a way that more carefully distinguishes between stiff and nonstiff dependence, such that in each time step only a linear system needs to be solved. These schemes provide an efficient tool for the numerical exploration of sediment formation and composition under a strongly degenerate polydisperse sedimentation model.     

Numerical schemes for kinematic flows with discontinuous flux

Spatially 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)     

流通量间断守恒高阶交通流模型及其数值模拟

在非均匀道路条件下，推广了各向异性守恒高阶交通流模型(CHO模型)，获得流通量间断CHO模型，并基于其Riemann不变量性质，运用局部简化方法及δ映射算法，设计了求解流通量间断CHO模型的一阶Godunov、EO(Engquist-Osher)和LF(Lax-Friedrichs)等数值格式.通过数值模拟表明流通量间断CHO模型是合理有效的，它可以描述平衡态和非平衡态交通流，相对于流通量间断LWR(Lighthill-Whitham-Richards)模型，其能更好地刻画实际交通现象.     

On entropy stable schemes for degenerate parabolic multispecies kinematic flow models

Entropy stable schemes for the numerical solution of initial value problems of nonlinear, possibly strongly degenerate systems of convection–diffusion equations were recently proposed in Jerez and Parés's study. These schemes extend the theoretical framework of Tadmor's study to convection–diffusion systems. They arise from entropy conservative schemes by adding a small amount of viscosity to avoid spurious oscillations. The main condition for feasibility of entropy conservative or stable schemes for a given model is that the corresponding first‐order system of conservation laws possesses a convex entropy function and corresponding entropy flux, and that the diffusion matrix multiplied by the inverse of the Hessian of the entropy is positive semidefinite. As a new contribution, it is demonstrated in the present work, first, that these schemes can naturally be extended to initial‐boundary value problems with zero‐flux boundary conditions in one space dimension, including an explicit bound on the growth of the total entropy. Second, it is shown that these assumptions are satisfied by certain diffusively corrected multiclass kinematic flow models of arbitrary size that describe traffic flow or the settling of dispersions and emulsions, where the latter application gives rise to zero‐flux boundary conditions. Numerical examples illustrate the behavior and accuracy of entropy stable schemes for these applications.     

Analytical studies on the instabilities of heterogeneous intelligent traffic flow

It has been widely reported in literature that a small perturbation in traffic flow such as a sudden deceleration of a vehicle could lead to the formation of traffic jams without a clear bottleneck. These traffic jams are usually related to instabilities in traffic flow. The applications of intelligent traffic systems are a potential solution to reduce the amplitude or to eliminate the formation of such traffic instabilities. A lot of research has been conducted to theoretically study the effect of intelligent vehicles, for example adaptive cruise control vehicles, using either computer simulation or analytical method. However, most current analytical research has only applied to single class traffic flow. To this end, the main topic of this paper is to perform a linear stability analysis to find the stability threshold of heterogeneous traffic flow using microscopic models, particularly the effect of intelligent vehicles on heterogeneous (or multi-class) traffic flow instabilities. The analytical results will show how intelligent vehicle percentages affect the stability of multi-class traffic flow.     

Integrated modelling and numerical treatment of freeway flow

The effectiveness of macroscopic dynamic freeway flow models at both uninterrupted and interrupted flow conditions is tested. Model implementation is made by finite difference methods developed here for solving the system's governing equations. These schemes are more effective than existing numerical methods, particularly when generation terms are introduced. The modelling alternatives and numerical solution algorithms are compared by employing a data base generated through microscopic simulation. Despite the effectiveness of the proposed numerical treatments, substantial deviations from the data at interrupted flows are still noticeable. In order to improve performance when flow is interrupted, we develop a modelling methodology that takes into account the ramp-freeway interactions so that all freeway components are treated as a system. We show that the coupling effects of the merging traffic streams are significant. Finally, the incremental benefits of using the more sophisticated high-order continuum models are assessed.     

A probabilistic model of thermal explosion in polydisperse fuel spray

This work is concerned with an analysis of polydisperse spray droplets distribution on the thermal explosion processes. In many engineering applications it is usual to relate to the practical polydisperse spray as a monodisperse spray. The Sauter Mean Diameter (SMD) and its variations are frequently used for this purpose [13]. The SMD and its modifications depend only on “integral” characterization of polydisperse sprays and can be the same for very different types of polydisperse spray distributions.The current work presents a new, simplified model of the thermal explosion in a combustible gaseous mixture containing vaporizing fuel droplets of different radii (polydisperse). The polydispersity is modeled using a probability density function (PDF) that corresponds to the initial distribution of fuel droplets size. This approximation of polydisperse spray is more accurate than the traditional ‘parcel’ approximation and permits an analytical treatment of the simplified model. Since the system of the governing equations represents a multi-scale problem, the method of invariant (integral) manifolds is applied.An explicit expression of the critical condition for thermal explosion limit is derived analytically. Numerical simulations demonstrate an essential dependence of these thermal explosion conditions on the PDF type and represent a natural generalization of the thermal explosion conditions of the classical Semenov theory.     

Effective bandwidths: Call admission,traffic policing and filtering for ATM networks

In this paper we review and extend the effective bandwidth results of Kelly [28], and Kesidis, Walrand and Chang [29, 6]. These results provide a framework for call admission schemes which are sensitive to constraints on the mean delay or the tail distribution of the workload in buffered queues. We present results which are valid for a wide variety of traffic streams and discuss their applicability for traffic management in ATM networks. We discuss the impact of traffic policing schemes, such as thresholding and filtering, on the effective bandwidth of sources. Finally we discuss effective bandwidth results for Brownian traffic models for which explicit results reveal the interaction arising in finite buffers.     

Discretizations of nonlinear differential equations using explicit nonstandard methods

In a recent paper, Chen and Solis investigated the appearance of spurious solutions when first-order ODEs are discretized using Runge–Kutta schemes. They concluded that the reliability of the numerical solutions to a particular ODE could be verified only by constructing several discrete models and comparing their numerical results with the known properties of the exact solutions. We demonstrate that by using nonstandard schemes, all the difficulties found by Chen and Solis can be eliminated, and that qualitatively correct numerical solutions are obtained for all values of the step size. We illustrate these issues by applying nonstandard finite-difference techniques to the logistic, sine, cubic, and Monod equations.     

On mathematical models and numerical simulation of the fluidization of polydisperse suspensions

The well-known Masliyah–Lockett–Bassoon (MLB) model for sedimentation of small particles is extended to fluidization of polydisperse suspensions. For N particle species that differ in size and density, this model leads to a first-order system of N conservation laws, which in general is of mixed (in the case N = 2, hyperbolic–elliptic) type. By a simple algebraic steady-state analysis, we derive necessary compatibility conditions on the size and density parameters that admit the formation of stationary fluidized beds. We then proceed to determine the composition of polydisperse fluidized beds of given compatible species by varying the fluidization velocity and the initial composition of the suspensions, and prove that, within the framework of the MLB model combined with the Richardson–Zaki formula, the constructed bidisperse beds always cause the equations to be hyperbolic. This means that these states are always predicted to be stable. The transient behaviour of the MLB model applied to fluidization is illustrated by three numerical examples, in which the system of conservation laws is solved for N = 2, N = 3 and N = 5, respectively. These examples illustrate the effects of bed expansion and layer inversion caused by successively increasing the applied fluidization velocity and show that the predicted fluidized states are indeed attained.     

A preliminary study on multiscale ELLAM schemes for transient advection‐diffusion equations

We study multiscale Eulerian–Lagrangian localized adjoint methods (MsELLAMs) for transient linear advection‐diffusion equations with oscillatory coefficients, which arise in mathematical models for describing flow and transport through heterogeneous porous media, composite material design, and other applications. Several MsELLAM schemes are presented and studied. Numerical experiments are presented to observe the numerical performance of these MsELLAM schemes. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

On models of polydisperse sedimentation with particle-size-specific hindered-settling factors

Polydisperse suspensions consist of particles differing in size or density that are dispersed in a viscous fluid. During sedimentation, the different particle species segregate and create areas of different composition. Spatially one-dimensional mathematical models of this process can be expressed as strongly coupled, nonlinear systems of first-order conservation laws. The solution of this system is the vector of volume fractions of each species as a function of depth and time, which will in general be discontinuous. It is well known that this system is strictly hyperbolic provided that the Masliyah–Lockett–Bassoon (MLB) flux vector is chosen, the particles have the same density, and the hindered-settling factor (a multiplicative algebraic expression appearing in the flux vector) does not depend on the particle size but is the same for all species. It is the purpose of this paper to prove that this hyperbolicity result remains valid in a fairly general class of cases where the hindered-settling factor does depend on particle size. This includes the common power-law type hindered-settling factor in which the exponent, sometimes called Richardson–Zaki exponent, is determined individually for each species, and is a decreasing function of particle size. The importance of this paper is two-fold: it proves stability for a class of polydisperse suspensions that was not covered in previous work, and it offers a new analysis of real data.     

Simulating vehicular traffic in a network using dynamic routing

We present a new numerical code which solves the Lighthill – Whitham model, the classic macroscopic model for vehicular traffic flow, in a network with multi-destinations. We use a high-resolution shock-capturing scheme with approximate Riemann solver to solve the partial differential equations of the Lighthill – Whitham theory. These schemes are very efficient, robust and moreover well adapted to simulations of traffic flows. We develop a theory of dynamic routing including a procedure for traffic flow assignment at junctions which reproduces the correct propagation of irregularities and ensures at the same time conservation of the number of vehicles.     

Some techniques for improving the resolution of finite difference component-wise WENO schemes for polydisperse sedimentation models

Polydisperse sedimentation models can be described by a system of conservation laws for the concentration of each species of solids. Some of these models, as the Masliyah–Locket–Bassoon model, can be proven to be hyperbolic, but its full characteristic structure cannot be computed in closed form. Component-wise finite difference WENO schemes may be used in these cases, but these schemes suffer from an excessive diffusion and may present spurious oscillations near shocks. In this work we propose to use a flux-splitting that prescribes less numerical viscosity for component-wise finite difference WENO schemes. We compare this technique with others to alleviate the diffusion and oscillatory behavior of the solutions obtained with component-wise finite difference WENO methods.     

Euler-type schemes for weakly coupled forward-backward stochastic differential equations and optimal convergence analysis

We introduce a new Euler-type scheme and its iterative algorithm for solving weakly coupled forward-backward stochastic differential equations (FBSDEs). Although the schemes share some common features with the ones proposed by C. Bender and J. Zhang [Ann. Appl. Probab., 2008, 18: 143–177], less computational work is needed for our method. For both our schemes and the ones proposed by Bender and Zhang, we rigorously obtain first-order error estimates, which improve the half-order error estimates of Bender and Zhang. Moreover, numerical tests are given to demonstrate the first-order accuracy of the schemes.     

An efficient numerical method for singular perturbation problems

In this paper, we propose a method for the numerical solution of singularly perturbed two-point boundary-value problems (BVPs). First, we develop two schemes to integrate initial–value problem (IVP) for system of two first-order differential equations, and then by using these schemes we solve the BVP. Precisely, we convert the second-order BVP into a system of first-order differential equations, and then apply the numerical schemes to obtain the solution. In order to get an initial condition for the system, we use the asymptotic approximate solution. Error estimates are derived and numerical examples are provided to illustrate the present method.     

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89-112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge-Kutta methods, Appl. Numer. Math. 58 (2008) 1675-1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed.A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.     

Discrete-analytical methods for the implementation of variational principles in environmental applications

A new method of constructing numerical schemes on the base of a variational principle for models including convection-diffusion operators is proposed. An original element is the use of analytical solutions of local adjoint problems formulated for the operators of convection-diffusion within the framework of the splitting technique. This results in numerical schemes which are absolutely stable, monotonic, transportive, and differentiable with respect to the state functions and parameters of the model. Artificial numerical diffusion is avoided due to the analytical solutions. The variational technique provides strong consistency between the numerical schemes of the main and adjoint problems. A theoretical study of the new class of schemes is given. The quality of the numerical approximations is demonstrated by an example of the non-linear Burgers equation. These new schemes enhance our variational methodology of environmental modelling. As one of the environmental applications, an inverse problem of risk assessment for Lake Baikal is presented.