We present two uniformly accurate numerical methods for discretizing the Zakharovsystem (ZS) with a dimensionless parameter 0< ε ≤ 1, which is inversely proportional to theacoustic speed. In the subsonic limit regime, i.e., 0< ε << 1, the solution of ZS propagates waves with O(ε)- andO(1)-wavelengths in time and space, respectively, and/or rapid outgoing initial layerswith speed O(1/ε) in space due to the singular perturbation of the wave operator in ZS and/or theincompatibility of the initial data. By adopting an asymptotic consistent formulation of ZS, wepresent a time-splitting exponential wave integrator (TS-EWI) method via applying a time-splittingtechnique and an exponential wave integrator for temporal derivatives in the nonlinear Schr ̈odingerequation and wave-type equation, respectively. By introducing a multiscale decomposition of ZS, wepropose a time-splitting multiscale time integrator (TS-MTI) method. Both methods are explicitand convergent exponentially in space for all kinds of initial data, which is uniformly for ε ∈ (0,1].The TS-EWI method is simpler to be implemented and it is only uniformly and optimally accuratein time for well-prepared initial data, while the TS-MTI method is uniformly and optimally accuratein time for any kind of initial data. Extensive numerical results are reported to show their efficiencyand accuracy, especially in the subsonic limit regime. Finally, the TS-MTI method is applied tostudy numerically convergence rates of ZS to its limiting models when ε→0+.

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We show that the median$m$($x$) in the gamma distribution with parameter$x$is a strictly convex function on the positive half-line.

In this paper, we review some recent progress made in [4, 5, 6] on finite difference schemes for viscous incompressible flows using vorticity formulation. The main purpose of this series of papers [4, 5, 6] is to resurrect the idea of using local vorticity boundary condition for unsteady calculation. The emphasis is on simplicity of the methods. Three main issues will be discussed: efficient time-stepping procedures and cell Reynolds number constraints, efficient methods in 3D on non-staggered grids and efficient high order methods using compact differencing.