A key difficulty in the analysis and numerical approximation of the shallow water equations is the nonconservative product of measures due to the gravitational force acting on a sloped bottom. Solutions may be nonunique, and numerical schemes are not only consistent discretizations of the shallow water equations, but they also determine how to model the physics. Our derivation is based on a subcell reconstruction using infinitesimal singular layers at the cell boundaries, as inspired by S. Noelle, Y. Xing, and C.-W. Shu [J. Comput. Phys., 226 (2007), pp. 29-58]. One key step is to separate the singular measures. Another aspect is the reconstruction of the solution variables in the singular layers. We study three reconstructions. The first leads to the well-known scheme of Audusse et al., [SIAM J. Sci. Comput., 25 (2004), pp. 2050-2065], which introduces the hydrostatic reconstruction. The second is a modification proposed in [T. Morales de Luna, M. J. Castro D´ıaz, and C. Par´es, Appl. Math. Comput., 219 (2013), pp. 9012-9032], which analyzes whether a wave has enough energy to overcome a step. The third is our new scheme, which borrows its structure from the wet-dry front. For a number of cases discussed in recent years, where water runs down a hill, Audusse’s scheme converges slowly or fails. Morales’ scheme gives a visible improvement. Both schemes are clearly outperformed by our new scheme.

In this paper, we discuss the waiting time phenomena for a class of non-Newtonian polytropic filtration equation with convection. According to the influence of the convection on the waiting time property, the convection is classified into three classes, that is, the strong convection, the mild convection and the weak convection. For different cases, the sufficient and necessary conditions on the initial data are given for the existence of waiting time respectively.

The zero dissipation limit of the compressible heat-conducting NavierStokes equations in the presence of the shock is investigated. It is shown that when the heat conduction coefficient and the viscosity coefficient satisfy = O (), = O (), c> 0, as 0 (see (1.3)), if the solution of the corresponding Euler equations is piecewise smooth with shock wave satisfying the Lax entropy condition, then there exists a smooth solution to the NavierStokes equations, which converges to the piecewise smooth shock solution of the Euler equations away from the shock discontinuity at a rate of . The proof is given by a combination of the energy estimates and the matched asymptotic analysis introduced in [3].

We study a biodegradation model for the time evolution of concentrations of contaminant, nutrient, and bacteria. The bacteria has a natural concentration which will increase when the nutrient reaches the substrate (contaminant). The growth utilizes nutrients and degrades the substrate. Eventually, such a process removes all the substrate and can be described by traveling wave solutions. The model consists of advection-reaction-diffusion equations for the substrate and nutrient concentrations and a rate equation (ODE) for the bacteria concentration. We first show the existence of approximate traveling wave solutions to an elliptically regularized system posed on a finite domain using degree theory and the elliptic maximum principle. To prove that the approximate solutions do not converge to trivial solutions, we construct comparison functions for each component and employ integral identities of the governing

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