In this paper, we extend the high order finite-difference method with subcell resolution (SR) for two-species stiff one-reaction models to multispecies and multireaction chemical reactive flows, which are significantly more difficult because of the multiple scales generated by different reactions. For reaction problems, when the reaction time scale is very small, the reaction zone scale is also small and the governing equations become very stiff. Wrong propagation speed of discontinuity may occur due to the underresolved numerical solution in both space and time. The present SR method for reactive Euler system is a fractional step method. In the convection step, any high order shock-capturing method can be used. In the reaction step, an ODE solver is applied but with certain computed flow variables in the shock region modified by the Harten subcell resolution idea. Several numerical examples of multispecies and multireaction reactive flows are performed in both one and two dimensions. Studies demonstrate that the SR method can capture the correct propagation speed of discontinuities in very coarse meshes.

We investigate the singularity formation of a 3D model that was recently proposed by Hou and Lei (2009) in [15] for axisymmetric 3D incompressible Navier–Stokes equations with swirl. The main difference between the 3D model of Hou and Lei and the reformulated 3D Navier–Stokes equations is that the convection term is neglected in the 3D model. This model shares many properties of the 3D incompressible Navier– Stokes equations. One of the main results of this paper is that we prove rigorously the finite time singularity formation of the 3D inviscid model for a class of initial boundary value problems with smooth initial data of finite energy. We also prove the global regularity of the 3D inviscid model for a class of small smooth initial data.

We observe that Sturm’s error bounds readily imply that for semidefinite feasibility problems, the method of alternating projections converges at a rate of O(k^{-1/(2^{d+1}-2)}), where d is the singularity degree of the problem—the minimal number of facial reduction iterations needed to induce Slater’s condition. Consequently, for almost all such problems (in the sense of Lebesgue measure), alternating projections converge at a worst-case rate of O(1/k^{0.5}).

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Thetransmissioneigenvalueproblem,besidesitscriticalroleininversescattering problems, deserves special interest of its own due to the fact that the corresponding differen- tial operator is neither elliptic nor self-adjoint. In this paper, we provide a spectral analysis and propose a novel iterative algorithm for the computation of a few positive real eigen- values and the corresponding eigenfunctions of the transmission eigenvalue problem. Based on approximation using continuous finite elements, we first derive an associated symmetric quadratic eigenvalue problem (QEP) for the transmission eigenvalue problem to eliminate the nonphysical zero eigenvalues while preserve all nonzero ones. In addition, the derived QEP enables us to consider more refined discretization to overcome the limitation on the number of degrees of freedom. We then transform the QEP to a parameterized symmet- ric definite generalized eigenvalue problem (GEP) and develop a secant-type iteration for solving the resulting GEPs. Moreover, we carry out spectral analysis for various existence intervals of desired positive real eigenvalues, since a few lowest positive real transmission eigenvalues are of practical interest in the estimation and the reconstruction of the index of refraction. Numerical experiments show that the proposed method can find those desired smallest positive real transmission eigenvalues accurately, efficiently, and robustly.