Abstract. In this paper, we develop a new discontinuous Galerkin (DG) finite element method for solving time dependent partial differential equations (PDEs) with higher order spatial derivatives. Unlike the traditional local discontinuous Galerkin (LDG) method, the method in this paper can be applied without introducing any auxiliary variables or rewriting the original equation into a larger system. Stability is ensured by a careful choice of interface numerical fluxes. The method can be designed for quite general nonlinear PDEs and we prove stability and give error estimates for a few representative classes of PDEs up to fifth order. Numerical examples show that our scheme attains the optimal (k + 1)-th order of accuracy when using piecewise k-th degree polynomials, under the condition that k + 1 is greater than or equal to the order of the equation.

In this paper, we study the superconvergence property for the discontinuous Galerkin (DG) and the local discontinuous Galerkin (LDG) methods, for solving one-dimensional time dependent linear conservation laws and convection-diffusion equations. We prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection-diffusion equations. The order of superconvergence for both cases is proved to be k + 3 2 when piecewise P k polynomials with k ≥ 1 are used. The proof is valid for arbitrary non-uniform regular meshes and for piecewise P k polynomials with arbitrary k ≥ 1, improving upon the results in [8, 9] in which the proof based on Fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise P 1 polynomials.

Abstract. Discontinuous Galerkin methods are developed for solving the Vlasov-Maxwell system, methods that are designed to be systematically as accurate as one wants with provable conservation of mass and possibly total energy. Such properties in general are hard to achieve within other numerical method frameworks for simulating the Vlasov-Maxwell system. The proposed scheme employs discontinuous Galerkin discretizations for both the Vlasov and the Maxwell equations, resulting in a consistent description of the distribution function and electromagnetic fields. It is proven, up to some boundary effects, that charge is conserved and the total energy can be preserved with suitable choices of the numerical flux for the Maxwell equations and the underlying approximation spaces. Error estimates are established for several flux choices. The scheme is tested on the streaming Weibel instability: the order of accuracy and conservation properties of the proposed method are verified.

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Abstract. In this paper, we develop a sparse grid discontinuous Galerkin (DG) scheme for transport equations and applied it to kinetic simulations. The method uses the weak formulations of traditional Runge-Kutta DG (RKDG) schemes for hyperbolic problems and is proven to be L 2 stable and convergent. A major advantage of the scheme lies in its low computational and storage cost due to the employed sparse finite element approximation space. This attractive feature is explored in simulating Vlasov and Boltzmann transport equations. Good performance in accuracy and conservation is verified by numerical tests in up to four dimensions.Key words. discontinuous Galerkin methods; sparse grid; high-dimensional transport equations; Vlasov equation; Boltzmann equation.1. Introduction. In this paper, we develop a sparse grid DG method for high-dimensional transport equations. High-dimensional transport problems are ubiquitous in science and engineering, and most evidently in kinetic simulations where it is necessary to track the evolution of probability density functions of particles. Deterministic kinetic simulations are very demanding due to the large computational and storage cost. To make the schemes more attractive comparing with the alternative probabilistic methods, an appealing approach is to explore the sparse grid techniques [6,15] with the aim of breaking the curse of dimensionality [4]. In the context of wavelets or sparse grid methods for kinetic transport equations, we mention the work of using wavelet-MRA methods for Vlasov equations [5], the combination technique for linear gyrokinetics [19], sparse adaptive finite element method [25], sparse discrete ordinates method [16] and sparse tensor spherical harmonics [17] for radiative transfer, among many others. This paper focuses on the DG method [13], which is a class of finite element methods using discontinuous approximation space for the numerical solution and the test functions. The RKDG scheme [14] developed in a series of papers for hyperbolic equations became very popular due to its provable convergence, excellent conservation properties and accommodation for adaptivity and parallel implementations. Recent years have seen great growth in the interest of applying DG methods to kinetic systems (see for example [3,18,20,9, 10]) because of the conservation properties and long time performance of the resulting simulations. However, the DG method is still deemed too costly in a realistic setting, often requiring more degrees of freedom than other high order numerical calculations.Recently, we developed a sparse grid DG method for high-dimensional elliptic problems [24]. A sparse DG finite element space has been constructed, reducing the degrees of freedom from the standardwhere h is the uniform mesh size in each dimension. The resulting scheme retains main properties of standard DG methods while making the computational cost tangebile for high-dimensional simulations. This motivates the current work for the transport equations, and we use kinetic problems as a...

This paper constitutes our initial effort in developing sparse grid discontinuous Galerkin (DG) methods for high-dimensional partial differential equations (PDEs). Over the past few decades, DG methods have gained popularity in many applications due to their distinctive features. However, they are often deemed too costly because of the large number of degrees of freedom of the approximation space, which are the main bottleneck for simulations in high dimensions. In this paper, we develop sparse grid DG methods for elliptic equations with the aim of breaking the curse of dimensionality. Using a hierarchical basis representation, we construct a sparse finite element approximation space, reducing the degrees of freedom from the standardproblems, where h is the uniform mesh size in each dimension. Our method, based on the interior penalty (IP) DG framework, can achieve accuracy of O(h k | log 2 h| d−1 ) in the energy norm, where k is the degree of polynomials used. Error estimates are provided and confirmed by numerical tests in multi-dimensions.

In this paper we consider Runge-Kutta discontinuous Galerkin (RKDG) schemes for Vlasov-Poisson systems that model collisionless plasmas. One-dimensional systems are emphasized. The RKDG method, originally devised to solve conservation laws, is seen to have excellent conservation properties, be readily designed for arbitrary order of accuracy, and capable of being used with a positivity-preserving limiter that guarantees positivity of the distribution functions. The RKDG solver for the Vlasov equation is the main focus, while the electric field is obtained through the classical representation by Green's function for the Poisson equation. A rigorous study of recurrence of the DG methods is presented by Fourier analysis, and the impact of different polynomial spaces and the positivity-preserving limiters on the quality of the solutions is ascertained. Several benchmark test problems, such as Landau damping, the two-stream instability, and the KEEN (Kinetic Electro static Electron Nonlinear) wave, are given.

In this paper, we present results of a discontinuous Galerkin (DG) scheme applied to deterministic computations of the transients for the Boltzmann-Poisson system describing electron transport in semiconductor devices. The collisional term models optical-phonon interactions which become dominant under strong energetic conditions corresponding to nano-scale active regions under applied bias. The proposed numerical technique is a finite element method using discontinuous piecewise polynomials as basis functions on unstructured meshes. It is applied to simulate hot electron transport in bulk silicon, in a silicon n + -n-n + diode and in a double gated 12nm MOSFET. Additionally, the obtained results are compared 1 Support from the Institute of Computational Engineering and Sciences and the University of Texas Austin is gratefully acknowledged.2 . Research supported by Italian PRIN 2006: Kinetic and continuum models for particle transport in gases and semiconductors: analytical and computational aspects. 5 to those of a high order WENO scheme simulation and DSMC (Discrete Simulation Monte Carlo) solvers.