This paper is concerned with the nonlinear stability of traveling wave solutions for a quasi-linear relaxation model with a nonconvex equilibrium flux. The study is motivated by and the results are applied to the well-known dynamic continuum traffic flow model, the Payne and Whitham (PW) model with a nonconcave fundamental diagram. The PW model is the first of its kind and it has been widely adopted by traffic engineers in the study of stability and instability phenomena of traffic flow. The traveling wave solutions are shown to be asymptotically stable under small disturbances and under the sub-characteristic condition using a weighted energy method. The analysis applies to both non-degenerate case and the degenerate case where the traveling wave has exponential decay rates at infinity and has an algebraic decay rate at infinity, respectively.

We provide a framework so that hyperbolic systems are endowed with a relative entropy identity. This allows a direct proof of convergence in the regime that the limiting solution is smooth.

The second-order Van-Leer MUSCL schemes are actually one of the most popular high order schemes for fluid dynamic computations. In the frame work of the Euler equations, we introduce a new slope limitation procedure to enforce the scheme to preserve the invariant region: namely the positiveness of both density and pressure as soon as the associated first order scheme does it. In addition, we obtain a second-order minimum principle on the specific entropy and second-order entropy inequalities. This new limitation is developed in the general framework of the MUSCL schemes and the choice of the numerical flux functions remains free. The proposed slope limitation can be applied to any change of variables and we do not impose the use of conservative variables in the piecewise linear reconstruction. Several examples are given in the framework of the primitive variables. Numerical 1D and 2D results are performed using several finite volume methods.

We present two new model equations for the unidirectional propagation of long waves in dispersive media for the specific purpose of modeling water waves. The derivation of the new equations uses a Pade(2,2) approximation of the phase velocity that arises in the linear water wave theory. Unlike the Korteweg-deVries (KdV) equation and similarly to the Benjamin-Bona-Mahony (BBM) equation, our models have a bounded dispersion relation. At the same time, the equations we propose provide the best approximation of the phase velocity for small wave numbers that can be obtained with third-order equations. We note that the new model equations can be transformed into previously studied models, such as the BBM and the Burgers-Poisson equations. It is therefore straightforward to establish the existence and uniqueness of solutions to the new equations. We also show that the distance between the solutions of one of the new equations, the KdV equation, and the BBM equation, is of the small order that is formally neglected by all models.

We prove a lower bound on the rate of realaxation to equililbrium in the H¹ norm for a thin film equation. We find a two stagerelaxation, with power law decay in an initial interval followed byexponential decay, at an essentially optimal rate, for large times. The waiting time until the expoential decay sets in is explicitly estimated.

A general framework for structure-preserving model reduction by Krylov subspace projection methods is developed. It not only matches as many moments as possible but also preserves substructures of importance in the coeficient matrices L,G,C, and B that define a dynamical system prescribed by the transfer function of the form H(s/) = L^*(G+sC)^-1B. Many existing structure-preserving model-order reduction methods for linear and second-order dynamical systems can be derived under this general framework. Furthermore, it also offers insights into the development of new structure-preserving model reduction methods.

We study the structure of stationary solutions to the Doi-Onsager equation with Maier-Saupe potential on the sphere, which arises in the modelling of rigid rod-like molecules of polymers. The stationary solutions are shown to be necessarily a set of axially symmetric functions, and a complete classification of parameters for phase transitions to these stationary solutions is obtained. It is shown that the number of stationary solutions hinges on whether the potential intensity crosses two critical values LOOK HERE!!!![PROPORTIONAL TO??] ₁ about 6.731393 and LOOK HERE!!!![PROPORTIONAL TO??]₂=7.5. Furthermore, we present explicit formulas for all stationary solutions.

A model for the transport of electrons in a semiconductor is considered where the electrons travel with saturation speed in the direction of the driving force computed self consistently from the Poisson equation. Since the velocity is discontinuous at zeroes of the driving force, an interpretation of the model in the distributional sense is not necessarily possible. For a spatially one-dimensional model existence of distributional solutions is shown by passing to the limit in a regularized problem corresponding to a scaled drift-diusion model with a velocity saturation assumption on the mobility. Several explicit solutions of the limiting problem are computed and compared to the results of numerical computations.

We consider high order Discontinuous Galerkin (DG) discretization for steady state problems. It will be demonstrated that using a high order DG scheme to discretize a problem may result in two types of solution overshoots. The oscillations of the first type are associated with smooth approximation of solution discontinuities. In addition, the oscillations may appear in steady state problems as a result of incorrect flux approximation near the flux extremum point.

An orthogonal discrete auditory transform (ODAT) from sound signal to spectrum is constructed by combining the auditory spreading matrix of Schroeder et. al. and the time one map of a discrete nonlocal Schrodinger equation. Thanks to the dispersive smoothing property of the Schrodinger evolution, ODAT spectrum is a smoother than that of the discrete Fourier transform (DFT) consistent with human audition. ODAT and DFT are compared in signal denoising tests with spectral thresholding method. The signals are noisy speech segments. ODAT outperforms DFT in signal to noise ratio (SNR) when the noise level is relatively high.