In this paper, we investigate a simple limiter using weighted essentially non-oscillatory (WENO) methodology for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving conservation laws, with the goal of obtaining a robust and high order limiting procedure to simultaneously achieve uniform high order accuracy and sharp, non-oscillatory shock transitions. The idea of this limiter is to reconstruct the entire polynomial, instead of reconstructing point values or moments in the classical WENO reconstructions. That is, the reconstruction polynomial on the target cell is a convex combination of polynomials on this cell and its neighboring cells and the nonlinear weights of the convex combination follow the classical WENO procedure. The main advantage of this limiter is its simplicity in implementation, especially for multi-dimensional meshes. Numerical results in one and two dimensions are provided to illustrate the behavior of this procedure.

This is an extension of our earlier work \cite{DS} in which a high order stable method was constructed for solving hyperbolic conservation laws on arbitrarily distributed point clouds. %The initial condition is given on such a point cloud, %and the algorithm solves for point values of the solution at %later time also on this point cloud. An algorithm of building a suitable polygonal mesh based on the random points was given and the traditional discontinuous Galerkin (DG) method was adopted on the constructed polygonal mesh. Numerical results in \cite{DS} show that the current scheme will generate spurious numerical oscillations when dealing with solutions containing strong shocks. In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for DG schemes on two-dimensional unstructured triangular meshes \cite{ZZSQ}, to our high order method on polygonal meshes. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the original method in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt is particularly simple to implement and will not harm the conservativeness and compactness of the original method. Moreover, we also extend the maximum-principle-satisfying limiter for the scalar case and the positivity-preserving limiter for the Euler system to our method. Numerical results for both scalar equations and Euler systems of compressible gas dynamics are provided to illustrate the good behavior of these limiters.

The purpose of this work is to incorporate numerically, in a discontinuous Galerkin (DG) solver of a Boltzmann-Poisson model for hot electron transport, an electronic conduction band whose values are obtained by the spherical averaging of the full band structure given by a local empirical pseudopotential method (EPM) around a local minimum of the conduction band for silicon, as a midpoint between a radial band model and an anisotropic full band, in order to provide a more accurate physical description of the electron group velocity and conduction energy band structure in a semiconductor. This gives a better quantitative description of the transport and collision phenomena that fundamentally define the behaviour of the Boltzmann - Poisson model for electron transport used in this work. The numerical values of the derivatives of this conduction energy band, needed for the description of the electron group velocity, are obtained by means of a cubic spline interpolation. The EPM-Boltzmann-Poisson transport with this spherically averaged EPM calculated energy surface is numerically simulated and compared to the output of traditional analytic band models such as the parabolic and Kane bands, numerically implemented too, for the case of 1D $n^+-n-n^+$ silicon diodes with $400nm$ and $50nm$ channels. Quantitative differences are observed in the kinetic moments related to the conduction energy band used, such as mean velocity, average energy, and electric current (momentum).

This is part II of our paper in which we propose and develop a dynamically bi-orthogonal method (DyBO) to study a class of time-dependent stochastic partial differential equations (SPDEs) whose solutions enjoy a low-dimensional structure. In part I of our paper [9], we derived the DyBO formulation and proposed numerical algorithms based on this formulation. Some important theoretical results regarding consistency and bi-orthogonality preservation were also established in the first part along with a range of numerical examples to illustrate the effectiveness of the DyBO method. In this paper, we focus on the computational complexity analysis and develop an effective adaptivity strategy to add or remove modes dynamically. Our complexity analysis shows that the ratio of computational complexities between the DyBO method and a generalized polynomial chaos method (gPC) is roughly of order O ((m/N p) 3) for a

In this paper, we introduce a general approach for proving optimal $L^2$ error estimates for the semi-discrete local discontinuous Galerkin (LDG) methods solving linear high order wave equations. The optimal order of error estimates hold not only for the solution itself but also for the auxiliary variables in the LDG method approximating the various order derivatives of the solution. Several examples including the one-dimensional third order wave equation, one-dimensional fifth order wave equation, and multi-dimensional Schr\"{o}dinger equation are explored to demonstrate this approach. The main idea is to derive energy stability for the various auxiliary variables in the LDG discretization, via using the scheme and its time derivatives with different test functions. Special projections are utilized to eliminate the jump terms at the cell boundaries in the error estimate in order to achieve the optimal order of accuracy.