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The Runge-Kutta discontinuous Galerkin (RKDG) method is a high order finite element method for solving hyperbolic conservation laws. It uses ideas from high resolution finite volume schemes, such as the exact or approximate Riemann solvers, total variation diminishing (TVD) Runge-Kutta time discretizations, and limiters. It has the advantage of flexibility in handling complicated geometry, h-p adaptivity, and efficiency of parallel implementation, and has been used successfully in many applications. However, the limiters used to control spurious oscillations in the presence of strong shocks are less robust than the strategies of essentially nonoscillatory (ENO) and weighted ENO (WENO) finite volume and finite difference methods. In this paper we investigate using WENO finite volume methodology as limiters for RKDG methods, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, nonoscillatory shock transition for RKDG methods. The traditional finite volume WENO framework based on cell averages is used to reconstruct point values of the solution at Gaussian-type points in those cells where limiting is deemed necessary, and the polynomial solutions in those cells are then rebuilt through numerical integration using these Gaussian points. Numerical results in one and two dimensions are provided to illustrate the behavior of this procedure.

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The aqueous dispersion of ZnO quantum dots (QDs) with strong blue emission (quantum yield of 76%) was synthesized through a simple solution route. The water stability of such QDs is provided by the hydroxyl groups on their surface, and the strong blue emission is suggested to arise from the formation of surface ZnO/oleic acid complexes. Under irradiation, these complexes are thought to absorb the excitation light with 3.54 eV and then generate the blue emission with 2.82 eV.

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In this paper we develop a Lax-Wendroff time discretization procedure for high order finite difference weighted essentially nonoscillatory schemes to solve hyperbolic conservation laws. This is an alternative method for time discretization to the popular TVD Runge-Kutta time discretizations. We explore the possibility in avoiding the local characteristic decompositions or even the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining nonoscillatory properties for problems with strong shocks. As a result, the Lax-Wendroff time discretization procedure is more cost effective than the Runge-Kutta time discretizations for certain problems including two-dimensional Euler systems of compressible gas dynamics.

In [SIAM J. Sci. Comput., 26 (2005), pp. 907-929], we initiated the study of using WENO (weighted essentially nonoscillatory) methodology as limiters for the RKDG (Runge-Kutta discontinuous Galerkin) methods. The idea is to first identify "troubled cells," namely, those cells where limiting might be needed, then to abandon all moments in those cells except the cell averages and reconstruct those moments from the information of neighboring cells using a WENO methodology. This technique works quite well in our one-and two-dimensional test problems [SIAM J. Sci. Comput., 26 (2005), pp. 907-929] and in the follow-up work where more compact Hermite WENO methodology is used in the troubled cells. In these works we used the classical minmod-type TVB (total variation bounded) limiters to identify the troubled cells; that is, whenever the minmod limiter attempts to change the slope, the cell is declared to be a troubled cell. This troubled-cell indicator has a TVB parameter M to tune and may identify more cells than necessary as troubled cells when M is not chosen adequately, making the method costlier than necessary. In this paper we systematically investigate and compare a few different limiter strategies as troubled-cell indicators with an objective of obtaining the most efficient and reliable troubled-cell indicators to save computational cost.