The computation of compressible flows becomes more challenging when the Mach number has different orders of magnitude. When the Mach number is of order one, modern shock capturing methods are able to capture shocks and other complex structures with high numerical resolutions. However, if the Mach number is small, the acoustic waves lead to stiffness in time and exces-sively large numerical viscosity, thus demanding much smaller time step and mesh size than normally needed for incompressible flow simulation. In this paper, we develop an all-speed asymptotic preserving (AP) numerical scheme for the compressible isentropic Euler and Navier-Stokes equations that is uni-formly stable and accurate for all Mach numbers. Our idea is to split the system into two parts: one involves a slow, nonlinear and conservative hyper-bolic system adequate for the use of modern shock capturing methods, and the other a linear hyperbolic system which contains the stiff acoustic dynam-ics, to be solved implicitly. This implicit part is reformulated into a standard pressure Poisson projection system, and thus possesses sufficient structure for efficient fast Fourier transform solution techniques. In the zero Mach number limit, the scheme automatically becomes a projection method-like incompress-ible solver. We present numerical results in one and two dimensions in both compressible and incompressible regimes.

A new residual-based stabilized finite element method is analyzed for solving the Stokes equations in terms of velocity and pressure, where the $H^{−1}$ norm is introduced in the measurement of the residuals to obtain a symmetric positive definite (SPD) method. The $H^{−1}$ norm is computable and can be always easily realized offline by the continuous linear finite element solution or the preconditioned counterpart of the Poisson Dirichlet problem. Although the $H^{−1}$ norm is computed in the linear element space, no matter what the finite element spaces for the velocity and the pressure are, optimal error bounds can be established when using continuous finite element pairs $R_l$−$R_m$ for velocity and pressure for any $l, m\ge 1$. Numerical experiments are performed to confirm the theoretical results obtained.

We review some important ideas in the design and analysis of robust overlapping domain decomposition algorithms for high-contrast multiscale problems.

Atmospheric temperature is one of the most important climate variables. This observational study presents detailed descriptions of the temperature variability imprinted in the 9-year brightness temperature data acquired by the Advanced Microwave Sounding Unit- Instrument A (AMSU-A) aboard Aqua since September 2002 over tropical oceans. A non-linear, adaptive method called the Ensemble Joint Multiple Extraction has been employed to extract the principal modes of variability in the AMSU-A/Aqua data. The semi-annual, annual, quasibiennial oscillation (QBO) modes and QBO–annual beat in the troposphere and the stratosphere have been successfully recovered. The modulation by the El Nin˜o/Southern oscillation (ENSO) in the troposphere was found and correlates well with the Multivariate ENSO Index. The longterm variations during 2002–2011 reveal a cooling trend (-0.5 K/decade at 10 hPa) in the tropical stratosphere; the trend below the tropical tropopause is not statistically significant due to the length of our data. A new tropospheric near-annual mode (period *1.6 years) was also revealed in the troposphere, whose existence was confirmed using National Centers for Environmental Prediction Reanalysis air temperature data. The near-annual mode in the troposphere is found to prevail in the eastern Pacific region and is coherent with a near-annual mode in the observed sea surface temperature over the Warm Pool region that has previously been reported. It remains a challenge for climate models to simulate the trends and principal modes of natural variability reported in this work.

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