In this paper, a new type of finite difference Hermite weighted essentially non-oscillatory (HWENO) schemes are constructed for solving Hamilton-Jacobi (HJ) equations. Point values of both the solution and its first derivatives are used in the HWENO reconstruction and evolved via time advancing. While the evolution of the solution is still through the classical numerical fluxes to ensure convergence to weak solutions, the evolution of the first derivatives of the solution is through a simple dimension-by-dimension non-conservative procedure to gain efficiency. The main advantages of this new scheme include its compactness in the spatial field and its simplicity in the reconstructions. Extensive numerical experiments in one and two dimensional cases are performed to verify the accuracy, high resolution and efficiency of this new scheme.

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Differential optical absorption spectroscopy (DOAS) is a powerful tool for detecting and quantifying trace gases in atmospheric chemistry\cite {Platt_Stutz08}. DOAS spectra consist of a linear combination of complex multi-peak multi-scale structures. Most DOAS analysis routines in use today are based on least squares techniques, for example, the approach developed in the 1970s uses polynomial fits to remove a slowly varying background, and known reference spectra to retrieve the identity and concentrations of reference gases. An open problem is to identify unknown gases in the fitting residuals for complex atmospheric mixtures.

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.

In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes, to the correction procedure via reconstruction (CPR) framework for solving conservation laws. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt in this paper is particularly simple to implement and will not harm the conservativeness of the CPR framework. Also, it uses information only from the target cell and its immediate neighbors, thus can maintain the compactness of the CPR framework. Since the CPR framework with the WENO limiter does not in general preserve positivity of the solution, we also extend the positivity preserving limiters to the CPR framework. Numerical results in one and two dimensions are provided to illustrate the good behavior of this procedure.