We analyze the central discontinuous Galerkin (DG) method for time-dependent linear conservation laws. In one dimension, optimal a priori $L^2$ error estimates of order $k+1$ are obtained for the semidiscrete scheme when piecewise polynomials of degree at most $k$ ($k\geq0$) are used on overlapping uniform meshes. %Our analysis is valid for both periodic boundary conditions and %for inflow-outflow boundary conditions. We then extend the analysis to multidimensions on uniform Cartesian meshes when piecewise tensor product polynomials are used on overlapping meshes. Numerical experiments are given to demonstrate the theoretical results.

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.

In this paper we investigate staggered discontinuous Galerkin method for the Helmholtz equation with large wave number on general quadrilateral and polygonal meshes. The method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids, and at the same time, is numerical flux free. Furthermore, it allows hanging nodes, which can be simply treated as additional vertices. By exploiting a modified duality argument, the stability and convergence can be proved under the condition that \kappa h is sufficiently small, where \kappa h is the wave number and \kappa h is the mesh size. Error estimates for both the scalar and vector variables in \kappa h norm are established. Several numerical experiments are tested to verify our theoretical results and to present the capability of our method for capturing singular solutions.

In this paper, we develop and analyze a new multiscale discontinuous Galerkin (DG) method for one-dimensional stationary Schr\"{o}dinger equations with open boundary conditions \bo{which have highly oscillating solutions}. Our method uses a smaller finite element space than the WKB local DG (WKB-LDG) method while achieving the same order of accuracy with no resonance errors. We prove that the DG approximation converges optimally with respect to the mesh size $h$ in $L^2$ norm without the typical constraint \bo{that $h$ has to be smaller than the wave length}. Numerical experiments were carried out to verify the second order optimal convergence rate of the method and to demonstrate its ability to capture oscillating solutions on coarse meshes in the applications to Schr\"{o}dinger equations.

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