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Abstract.This is the second paper in a series in which we construct and analyze a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws ut + J^ =1(fi(u))Xi = 0. In this paper we present a general framework of the methods, up to any order of formal accuracy, using scalar one-dimensional initial value and initial-boundary problems as models. In these cases we prove TVBM (total variation bounded in the means), TVB, and convergence of the schemes. Numerical results using these methods are also given. Extensions to systems and/or higher dimensions will appear in future papers.

Hamilton-Jacobi (H-J) equations are frequently encountered in applications, e.g., in control theory and differential games. H-J equations are closely related to hyperbolic conservation laws--in one space dimension the former is simply the integrated version of the latter. Similarity also exists for the multidimensional case, and this is helpful in the design of difference approximations. In this paper high-order essentially nonoscillatory (ENO) schemes for H-J equations are investigated, which yield uniform high-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives. The ENO scheme construction procedure is adapted from that for hyperbolic conservation laws. The schemes are numerically tested on a variety of one-dimensional and two-dimensional problems, including a problem related to control optimization, and high-order accuracy in smooth regions, good resolution of discontinuities in the derivatives, and convergence to viscosity solutions are observed.

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High order accurate weighted essentially non-oscillatory (WENO) schemes are relatively new but have gained rapid popularity in numerical solutions of hyperbolic partial differential equations and other convection dominated problems. The main advantage of such schemes is their capability to achieve arbitrarily high order formal accuracy in smooth regions while maintaining stable, non-oscillatory and sharp discontinuity transitions. The schemes are thus especially suitable for problems containing both strong discontinuities and complex smooth solution features. WENO schemes are robust and do not require the users to tune parameters, thus they are very convenient to use for practitioners. In this paper we review the history and basic formulation of WENO schemes, outline the main ideas in using WENO schemes to solve various hyperbolic partial differential equations and other convection dominated problems, and present a collected sample of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, and computational biology. Finally, we mention a few topics currently being investigated about WENO schemes.

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When an optical dipole is in proximity to a metallic substrate, it can emit light into both far field photons and SPPs. Far-field emission can be measured directly via top-down optical microscopy, whereas SPP emission can be detected by converting SPPs into far-field light via engineered out-coupling structures (Fig. 1a). On a single-crystal silver film, our metal of choice due to its low loss 11 , SPPs are strongly polarized in the out-of-plane (z) direction in the visible frequency range (see Supplementary Information). Consequently, the emission rate into SPPs for an out-of-plane dipole can be as high as 30 times larger than that of an in-plane dipole (Figs. 1b-d, for details of the analysis see Supplementary Information). At the same time, far-field emission of an in-plane dipole is strongly suppressed (Figs. 1b and d) because the in-plane electric field is close to zero near the silver surface. We note that when a point dipole is close to a metal 12 , non-radiative recombination due to ohmic loss can be the dominant decay mechanism.Remarkably, for delocalized excitons in quantum wells and 2D materials, quenching of exciton luminescence by ohmic loss is significantly reduced, even when they are placed 10 nm above a silver surface ( see [ 13 ] and Supplementary Fig. 1 and discussion). Combined together, the net effect of a nearby silver surface is significantly enhanced (suppressed) emission of an out-ofplane (in-plane) dipole into SPPs (far field). (Fig. 1a). The spacing between the monolayer TMD and the silver surface is determined by the bottom hBN thickness, and can easily be controlled by varying hBN thickness. In our devices, the typical spacing is on the order of ten nanometers.Excitons are created using off-resonant 660-nm laser excitation, and the PL spectra are voltages. We normalize both FF and SPP-PL spectra using the intensity of a charged exciton peak X T because it is known to involve a purely in-plane transition dipole moment 16 . The ratio of SPP-PL intensity to the FF-PL intensity after the normalization provides a direct measure of the orientation of the transition dipole for each luminescent species: the unity ratio represents a purely in-plane dipole, while a value larger than one indicates that the transition dipole has some out-of-plane components. Based on our theoretical calculations presented in Fig. 1d and Supplementary Fig. 4, an optical transition with a purely out-of-plane transition dipole should have a normalized coupling ratio of 7 in our device geometry. The experimental results for X D yield a value of 16: this discrepancy between theory and experiment is likely due to small, yet non-negligible absorption of SPPs by charged excitons as they propagate through the WSe 2 , which increases the apparent coupling ratio of X D after normalization (see Supplementary Fig. 5). Indeed, when SPPs propagate through a minimal distance within WSe 2 ( Supplementary Fig. 6), the normalized coupling ratio determined by experiment is close to 7, in good agreement with the theoretical calc...