Solving wave propagation problems within heterogeneous media has been of great interest and has a wide range of applications in physics and engineering. The design of numerical methods for such general wave propagation problems is challenging because the energy conserving property has to be incorporated in the numerical algorithms in order to minimize the phase or shape errors after long time integration. In this paper, we focus on multi-dimensional wave problems and consider linear second-order wave equation in heterogeneous media. We develop and analyze an LDG method, in which numerical fluxes are carefully designed to maintain the energy preserving property and accuracy. Compatible high order energy conserving time integrators are also proposed. The optimal error estimates and the energy conserving property are proved for the semi-discrete methods. Our numerical experiments demonstrate optimal rates of convergence, and show that the errors of the numerical solutions do not grow significantly in time due to the energy conserving property.

Previous work has shown that in a two-dimensional periodic medium under focusing or defocusing cubic nonlinearities, gap solitons in the form of low-amplitude and slowly modulated single-Bloch-wave packets can bifurcate out from the edges of Bloch bands. In this paper, linear stability properties of these gap solitons near band edges are determined both analytically and numerically. Through asymptotic analysis, it is shown that these gap solitons are linearly unstable if the slope of their power curve at the band edge has the opposite sign of nonlinearity
here focusing nonlinearity is said to have a positive sign, and defocusing nonlinearity to have a negative sign. An equivalent condition for linear instability is that the power of the gap solitons near the band edge is lower than the limit power value on the band edge. Through numerical computations of the power curves, it is found that this condition is always satisfied, thus two-dimensional gap solitons near band edges are linearly unstable. The analytical formula for the unstable eigenvalue of gap solitons near band edges is also asymptotically derived. It is shown that this unstable eigenvalue is proportional to the cubic power of the soliton’s amplitude, and it induces width instabilities of gap solitons. A comparison between this analytical eigenvalue formula and numerically computed eigenvalues shows excellent agreement.

This paper is concerned with superconvergence properties of discontinuous Galerkin (DG) methods for 2-D linear hyperbolic conservation laws over rectangular meshes when upwind fluxes are used. We prove, under some suitable initial and boundary discretizations, the ($2k+1$)-th order superconvergence rate of the DG approximation at the downwind points and for the cell averages, when piecewise tensor-product polynomials of degree $k$ are used. Moreover, we prove that the gradient of the DG solution is superconvergent with a rate of ($k+1$)-th order at all interior left Radau points; and the function value approximation is superconvergent at all right Radau points with a rate of ($k+2$)-th order. Numerical experiments indicate that the aforementioned superconvergence rates are sharp.

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.

In a previous work [P. Crispel, P. Degond, and M.-H. Vignal, J. Comput. Phys., 223 (2007), pp. 208–234], a new numerical discretization of the Euler–Poisson system was proposed. This scheme is “asymptotic preserving” in the quasineutral limit (i.e., when the Debye length $\varepsilon$ tends to zero), which means that it becomes consistent with the limit model when $\varepsilon \to 0$. In the present work, we show that the stability domain of the present scheme is independent of $\varepsilon$. This stability analysis is performed on the Fourier transformed (with respect to the space variable) linearized system. We show that the stability property is more robust when a space-decentered scheme is used (which brings in some numerical dissipation) rather than a space-centered scheme. The linearization is first performed about a zero mean velocity and then about a nonzero mean velocity. At the various stages of the analysis, our scheme is compared with more classical schemes and its improved stability property is outlined. The analysis of a fully discrete (in space and time) version of the scheme is also given. Finally, some considerations about a model nonlinear problem, the Burgers–Poisson problem, are also discussed.