Real‐time road traffic forecasting using regime‐switching space‐time models and adaptive LASSO

Smart transportation technologies require real‐time traffic prediction to be both fast and scalable to full urban networks. We discuss a method that is able to meet this challenge while accounting for nonlinear traffic dynamics and space‐time dependencies of traffic variables. Nonlinearity is taken into account by a union of non‐overlapping linear regimes characterized by a sequence of temporal thresholds. In each regime, for each measurement location, a penalized estimation scheme, namely the adaptive absolute shrinkage and selection operator (LASSO), is implemented to perform model selection and coefficient estimation simultaneously. Both the robust to outliers least absolute deviation estimates and conventional LASSO estimates are considered. The methodology is illustrated on 5‐minute average speed data from three highway networks. Copyright © 2012 John Wiley & Sons, Ltd.     

High‐order finite difference schemes for numerical solutions of the generalized Burgers–Huxley equation

In this article, up to tenth‐order finite difference schemes are proposed to solve the generalized Burgers–Huxley equation. The schemes based on high‐order differences are presented using Taylor series expansion. To establish the numerical solutions of the corresponding equation, the high‐order schemes in space and a fourth‐order Runge‐Kutta scheme in time have been combined. Numerical experiments have been conducted to demonstrate the high‐order accuracy of the current algorithms with relatively minimal computational effort. The results showed that use of the present approaches in the simulation is very applicable for the solution of the generalized Burgers–Huxley equation. The current results are also seen to be more accurate than some results given in the literature. The proposed algorithms are seen to be very good alternatives to existing approaches for such physical applications. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1313‐1326, 2011     

Minimum time‐step criteria for the Galerkin finite element methods applied to one‐dimensional parabolic partial differential equations

The finite element method has been well established for numerically solving parabolic partial differential equations (PDEs). Also it is well known that a too large time step should not be chosen in order to obtain a stable and accurate numerical solution. In this article, accuracy analysis shows that a too small time step should not be chosen either for some time‐stepping schemes. Otherwise, the accuracy of the numerical solution cannot be improved or can even be worsened in some cases. Furthermore, the so‐called minimum time step criteria are established for the Crank‐Nicolson scheme, the Galerkin‐time scheme, and the backward‐difference scheme used in the temporal discretization. For the forward‐difference scheme, no minimum time step exists as far as the accuracy is concerned. In the accuracy analysis, no specific initial and boundary conditions are invoked so that such established criteria can be applied to the parabolic PDEs subject to any initial and boundary conditions. These minimum time step criteria are verified in a series of numerical experiments for a one‐dimensional transient field problem with a known analytical solution. The minimum time step criteria developed in this study are useful for choosing appropriate time steps in numerical simulations of practical engineering problems. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

A multilevel finite element method in space‐time for the Navier‐Stokes problem

A multilevel finite element method in space‐time for the two‐dimensional nonstationary Navier‐Stokes problem is considered. The method is a multi‐scale method in which the fully nonlinear Navier‐Stokes problem is only solved on a single coarsest space‐time mesh; subsequent approximations are generated on a succession of refined space‐time meshes by solving a linearized Navier‐Stokes problem about the solution on the previous level. The a priori estimates and error analysis are also presented for the J‐level finite element method. We demonstrate theoretically that for an appropriate choice of space and time mesh widths: h_j ～ h, k_j ～ k, j = 2, …, J, the J‐level finite element method in space‐time provides the same accuracy as the one‐level method in space‐time in which the fully nonlinear Navier‐Stokes problem is solved on a final finest space‐time mesh. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

A space‐time mixed‐hybrid finite element method for the damped wave equation

A space‐time finite element method is introduced to solve the linear damped wave equation. The scheme is constructed in the framework of the mixed‐hybrid finite element methods, and where an original conforming approximation of H(div;Ω) is used, the latter permits us to obtain an upwind scheme in time. We establish the link between the nonstandard finite difference scheme recently introduced by Mickens and Jordan and the scheme proposed. In this regard, two approaches are considered and in particular we employ a formulation allowing the solution to be marched in time, i.e., one only needs to consider one time increment at a time. Numerical results are presented and compared with the analytical solution illustrating good performance of the present method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

A BGK‐penalization‐based asymptotic‐preserving scheme for the multispecies Boltzmann equation

An asymptotic‐preserving (AP) scheme is efficient in solving multiscale problems where kinetic and hydrodynamic regimes coexist. In this article, we extend the BGK‐penalization‐based AP scheme, originally introduced by Filbet and Jin for the single species Boltzmann equation (Filbet and Jin, J Comput Phys 229 (2010) 7625–7648), to its multispecies counterpart. For the multispecies Boltzmann equation, the new difficulties arise due to: (1) the breaking down of the conservation laws for each species and (2) different convergence rates to equilibria for different species in disparate masses systems. To resolve these issues, we find a suitable penalty function—the local Maxwellian that is based on the mean velocity and mean temperature and justify various asymptotic properties of this method. This AP scheme does not contain any nonlinear nonlocal implicit solver, yet it can capture the fluid dynamic limit with time step and mesh size independent of the Knudsen number. Numerical examples demonstrate the correct asymptotic‐behavior of the scheme. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

Relaxation‐time limit of the three‐dimensional hydrodynamic model with boundary effects

In this paper, we study three‐dimensional (3D) unipolar and bipolar hydrodynamic models and corresponding drift‐diffusion models from semiconductor devices on bounded domain. Based on the asymptotic behavior of the solutions to the initial boundary value problems with slip boundary condition, we investigate the relation between the 3D hydrodynamic semiconductor models and the corresponding drift‐diffusion models. That is, we discuss the relation‐time limit from the 3D hydrodynamic semiconductor models to the corresponding drift‐diffusion models by comparing the large‐time behavior of these two models. These results can be showed by energy arguments. Copyrightcopyright 2011 John Wiley & Sons, Ltd.     

Convergence and error bound analysis for the space‐time CESE method

In this work, we study the convergence behavior of a recently developed space‐time conservation element and solution element method for solving conservation laws. In particular, we apply the method to a one‐dimensional time‐dependent convection‐diffusion equation possibly with high Peclet number. We prove that the scheme converges and we obtain an error bound. This method performs well even for strong convection dominance over diffusion with good long‐time accuracy. Numerical simulations are performed to verify the results. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 64–78, 2001     

A priori error estimates of an extrapolated space‐time discontinuous galerkin method for nonlinear convection‐diffusion problems

We deal with the numerical solution of a scalar nonstationary nonlinear convection‐diffusion equation. We employ a combination of the discontinuous Galerkin finite element (DGFE) method for the space as well as time discretization. The linear diffusive and penalty terms are treated implicitly whereas the nonlinear convective term is treated by a special higher order explicit extrapolation from the previous time step, which leads to the necessity to solve only a linear algebraic problem at each time step. We analyse this scheme and derive a priori asymptotic error estimates in the L^∞(L²) –norm and the L²(H¹) –seminorm with respect to the mesh size h and time step τ. Finally, we present an efficient solution strategy and numerical examples verifying the theoretical results. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1456–1482, 2010     

A second‐order linearized difference scheme on nonuniform meshes for nonlinear parabolic systems with Dirichlet boundary value conditions

A linearized three‐level difference scheme on nonuniform meshes is derived by the method of the reduction of order for the Dirichlet boundary value problem of the nonlinear parabolic systems. It is proved that the difference scheme is uniquely solvable and second order convergent in L_∞‐norm. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 638–652, 2003     

A parameter‐uniform scheme for the parabolic singularly perturbed problem with a delay in time

In this paper, a parameter‐uniform numerical scheme for the solution of singularly perturbed parabolic convection–diffusion problems with a delay in time defined on a rectangular domain is suggested. The presence of the small diffusion parameter ? leads to a parabolic right boundary layer. A collocation method consisting of cubic B ‐spline basis functions on an appropriate piecewise‐uniform mesh is used to discretize the system of ordinary differential equations obtained by using Rothe's method on an equidistant mesh in the temporal direction. The parameter‐uniform convergence of the method is shown by establishing the theoretical error bounds. The numerical results of the test problems validate the theoretical error bounds.     

A finite‐volume scheme for the multidimensional quantum drift‐diffusion model for semiconductors

A finite‐volume scheme for the stationary unipolar quantum drift‐diffusion equations for semiconductors in several space dimensions is analyzed. The model consists of a fourth‐order elliptic equation for the electron density, coupled to the Poisson equation for the electrostatic potential, with mixed Dirichlet‐Neumann boundary conditions. The numerical scheme is based on a Scharfetter‐Gummel type reformulation of the equations. The existence of a sequence of solutions to the discrete problem and its numerical convergence to a solution to the continuous model are shown. Moreover, some numerical examples in two space dimensions are presented. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1483–1510, 2011     

A non‐autonomous epidemic model with time delay and vaccination

In this paper, a non‐autonomous SIRVS epidemic model with time delay and vaccination is investigated. We assume that the vaccinated have a constant immunity period. Some new threshold conditions are obtained. These threshold conditions govern the extinction and permanence of the disease. When the model degenerates into the periodic or autonomous case, the corresponding basic reproduction number can be derived from these threshold conditions. Copyright © 2009 John Wiley & Sons, Ltd.     

A characteristic domain decomposition and space–time local refinement method for first‐order linear hyperbolic equations with interfaces

We develop a characteristic‐based domain decomposition and space–time local refinement method for first‐order linear hyperbolic equations. The method naturally incorporates various physical and numerical interfaces into its formulation and generates accurate numerical solutions even if large time‐steps are used. The method fully utilizes the transient and strongly local behavior of the solutions of hyperbolic equations and provides solutions with significantly improved accuracy and efficiency. Several numerical experiments are presented to illustrate the performance of the method and for comparison with other domain decomposition and local refinement schemes. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 1–28, 1999     

Fourth‐order compact scheme with local mesh refinement for option pricing in jump‐diffusion model

The value of a contingent claim under a jump‐diffusion process satisfies a partial integro‐differential equation. A fourth‐order compact finite difference scheme is applied to discretize the spatial variable of this equation. It is discretized in time by an implicit‐explicit method. Meanwhile, a local mesh refinement strategy is used for handling the nonsmooth payoff condition. Moreover, the numerical quadrature method is exploited to evaluate the jump integral term. It guarantees a Toeplitz‐like structure of the integral operator such that a fast algorithm is feasible. Numerical results show that this approach gives fourth‐order accuracy in space. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Chaotic fractional‐order model for muscular blood vessel and its control via fractional control scheme

This article studies the chaotic and complex behavior in a fractional‐order biomathematical model of a muscular blood vessel (MBV). It is shown that the fractional‐order MBV (FOMBV) model exhibits very complex and rich dynamics such as chaos. We show that the corresponding maximal Lyapunov exponent of the FOMBV system is positive which implies the existence of chaos. Strange attractors of the FOMBV model are depicted to validate the chaotic behavior of the system. We change the fractional order of the model and investigate the dynamics of the system. To suppress the chaotic behavior of the model, we propose a single input fractional finite‐time controller and prove its stability using the fractional Lyapunov theory. In addition, the effects of the model uncertainties and external disturbances are taken into account and a robust fractional finite‐time controller is constructed. The upper bound of the chaos suppression time is also given. Some computer simulations are presented to illustrate the findings of this article. © 2014 Wiley Periodicals, Inc. Complexity 20: 37–46, 2014     

A decoupling method with different subdomain time steps for the nonstationary stokes–darcy model

This report analyzes a multirate, decoupling algorithm, which allows different time steps in the fluid region and the porous region for the nonstationary Stokes–Darcy problem. The method presented requires only one, uncoupled Stokes and Darcy subphysics and subdomain solve per time step. Under a time step restriction of the form △t ≤ C (physical parameters) we prove stability and convergence of the method. Numerical tests are given to show the convergence result and demonstrate the computational efficiency of the partitioned method. They also show that in the expected case of greater fluid velocities in the free‐flow region than in the porous media region, allowing smaller time steps in the subregion with the faster velocities increases both accuracy and efficiency. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Fractional‐order switching type control law design for adaptive sliding mode technique of 3D fractional‐order nonlinear systems

In this article, an adaptive sliding mode technique based on a fractional‐order (FO) switching type control law is designed to guarantee robust stability for a class of uncertain three‐dimensional FO nonlinear systems with external disturbance. A novel FO switching type control law is proposed to ensure the existence of the sliding motion in finite time. Appropriate adaptive laws are shown to tackle the uncertainty and external disturbance. The calculation formula of the reaching time is analyzed and computed. The reachability analysis is visualized to show how to obtain a shorter reaching time. A stability criteria of the FO sliding mode dynamics is derived based on indirect approach to Lyapunov stability. Effectiveness of the proposed control scheme is illustrated through numerical simulations. © 2015 Wiley Periodicals, Inc. Complexity 21: 363–373, 2016     

A macroscopic model for an intermediate state between type‐I and type‐II superconductivity

A vectorial nonlocal and nonlinear parabolic problem on a bounded domain for an intermediate state between type‐I and type‐II superconductivity is proposed. The domain is for instance a multiband superconductor that combines the characteristics of both types. The nonlocal term is represented by a (space) convolution with a singular kernel arising in Eringen's model. The nonlinearity is coming from the power law relation by Rhyner. The well‐posedness of the problem is discussed under low regularity assumptions and the error estimate for a semi‐implicit time‐discrete scheme based on backward Euler approximation is established. In the proofs, the monotonicity methods and the Minty–Browder argument are used. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1551–1567, 2015