Global existence, singularities and ill-posedness for a nonlocal flux

In this paper we study a one-dimensional model equation with a nonlocal flux given by the Hilbert transform that is related with the complex inviscid Burgers equation. This equation arises in different contexts to characterize nonlocal and nonlinear behaviors. We show global existence, local existence, blow-up in finite time and ill-posedness depending on the sign of the initial data for classical solutions.     

Convergence of MPFA on triangulations and for Richards' equation

Spatial discretization of transport and transformation processes in porous media requires techniques that handle general geometry, discontinuous coefficients and are locally mass conservative. Multi‐point flux approximation (MPFA) methods are such techniques, and we will here discuss some formulations on triangular grids with further application to the nonlinear Richards equation. The MPFA methods will be rewritten to mixed form to derive stability conditions and error estimates. Several MPFA versions will be shown, and the versions will be discussed with respect to convergence, symmetry and robustness when the grids are rough. It will be shown that the behavior may be quite different for challenging cases of skewness and roughness of the simulation grids. Further, we apply the MPFA discretization approach for the Richards equation and derive new error estimates without extra regularity requirements. The analysis will be accompanied by numerical results for grids that are relevant for practical simulation. Copyright © 2008 John Wiley & Sons, Ltd.     

Numerical experiments with several variant WENO schemes for the Euler equations

Numerical experiments with several variants of the original weighted essentially non‐oscillatory (WENO) schemes (J. Comput. Phys. 1996; 126 :202–228) including anti‐diffusive flux corrections, the mapped WENO scheme, and modified smoothness indicator are tested for the Euler equations. The TVD Runge–Kutta explicit time‐integrating scheme is adopted for unsteady flow computations and lower–upper symmetric‐Gauss–Seidel (LU‐SGS) implicit method is employed for the computation of steady‐state solutions. A numerical flux of the variant WENO scheme in flux limiter form is presented, which consists of first‐order and high‐order fluxes and allows for a more flexible choice of low‐order schemes. Computations of unsteady oblique shock wave diffraction over a wedge and steady transonic flows over NACA 0012 and RAE 2822 airfoils are presented to test and compare the methods. Various aspects of the variant WENO methods including contact discontinuity sharpening and steady‐state convergence rate are examined. By using the WENO scheme with anti‐diffusive flux corrections, the present solutions indicate that good convergence rate can be achieved and high‐order accuracy is maintained and contact discontinuities are sharpened markedly as compared with the original WENO schemes on the same meshes. Copyright © 2008 John Wiley & Sons, Ltd.     

Impact of sideband harmonic pattern on losses in laminated non-oriented steel

Losses in laminated non-oriented steel under the effect of two high harmonic clusters, caused by voltage excitation typical of two-level pulse-with-modulated (PWM) DC-link and space-vector modulated (SVM) matrix converters, were analyzed. The predicting method proposed, which incorporates anisotropy of loss K_a and grain size g_s, describes the magnetizing process within the steel by the means of Poisson statistical distribution. Results are then compared to losses determined by Bertotti's model. The two methods confirmed that spreading of sideband harmonics in the kilohertz range can reduce harmonic losses by up to 40% at low power frequencies.     

Shock formation in a chemotaxis model

In this paper, we establish the existence of shock solutions for a simplified version of the Othmer–Stevens chemotaxis model (SIAM J. Appl. Math. 1997; 57 :1044–1081). The existence of these shock solutions was suggested by Levine and Sleeman (SIAM J. Appl. Math. 1997; 57 :683–730). Here, we consider the general Riemann problem and derive the shock curves in parameterized forms. By studying the travelling wave solutions, we examine the shock structure for the chemotaxis model and prove that the travelling wave speed is identical to the shock speed. Moreover, we explicitly derive an entropy–entropy flux pair to prove the uniqueness of the weak shock solutions. Some discussion is given for further study. Copyright © 2007 John Wiley & Sons, Ltd.     

Towards understanding the critical heat flux for industrial applications

Understanding CHF is of an upmost importance in many industries, especially in the design and operation of boilers, nuclear power plants, cryogenic systems, etc. Due to safety issues related to the nuclear power plants, and the adaptation of CHF as the limiting criterion of power generation, it is important to understand the mechanisms of CHF relevant to nuclear systems operation. Moreover, CHF is expected to occur during transients than steady-state conditions. Therefore, knowledge of transient CHF is of great importance for the safety evaluation of nuclear reactors under transient condition. In this paper, the existing CHF experimental and modeling studies are discussed in order to understand the phenomena leading to CHF. Also, the effect of transient conditions on CHF for nuclear fuels has been evaluated.     

Development and validation of a SUPG finite element scheme for the compressible Navier–Stokes equations using a modified inviscid flux discretization

This paper considers the streamline‐upwind Petrov–Galerkin (SUPG) method applied to the unsteady compressible Navier–Stokes equations in conservation‐variable form. The spatial discretization, including a modified approach for interpolating the inviscid flux terms in the SUPG finite element formulation, and the second‐order accurate time discretization are presented. The numerical method is discussed in detail. The performance of the algorithm is then investigated by considering inviscid flow past a circular cylinder. Validation of the finite element formulation via comparisons with experimental data for high‐Mach number perfect gas laminar flows is presented, with a specific focus on comparisons with experimentally measured skin friction and convective heat transfer on a 15° compression ramp. Copyright © 2007 John Wiley & Sons, Ltd.     

A Reaction-diffusion system with nonlinear absorption terms and boundary flux

This paper deals with a reaction-diffusion system with nonlinear absorption terms and boundary flux. As results of interactions among the six nonlinear terms in the system, some sufficient conditions on global existence and finite time blow-up of the solutions are described via all the six nonlinear exponents appearing in the six nonlinear terms. In addition, we also show the influence of the coefficients of the absorption terms as well as the geometry of the domain to the global existence and finite time blow-up of the solutions for some cases. At last, some numerical results are given.     

关于一类HJ方程的Godunov通量的一点注记

本文对[1]中关于一类HJ方程的Godunov通量的命题作出改进,并给出新的证明.     

Efficient construction of high‐resolution TVD conservative schemes for equations with source terms: application to shallow water flows

High‐resolution total variation diminishing (TVD) schemes are widely used for the numerical approximation of hyperbolic conservation laws. Their extension to equations with source terms involving spatial derivatives is not obvious. In this work, efficient ways of constructing conservative schemes from the conservative, non‐conservative or characteristic form of the equations are described in detail. An upwind, as opposed to a pointwise, treatment of the source terms is adopted here, and a new technique is proposed in which source terms are included in the flux limiter functions to get a complete second‐order compact scheme. A new correction to fix the entropy problem is also presented and a robust treatment of the boundary conditions according to the discretization used is stated. Copyright © 2001 John Wiley & Sons, Ltd.