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A class of high order central compact schemes with spectral-like resolution are designed for the computational aeroacousitcs (CAA). The schemes have the features of high order, high resolution, low dissipation and the ability to capture strong shock wave in flow field, which are perfect methods for computational aeroacoustics. Typical problems are solved through direct numerical simulation, including the merging process of two co-rotating Gaussian vortices, the interaction between an oblique shock wave and a shear layer and cavity flow. The mechanisms of noise generation are studied.

In this presentation we give an overview of some recent progress in high order finite difference and finite volume weighted essentially non-oscillatory (WENO) and finite element discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws and in general convection dominated PDEs, such as convection diffusion equations with small diffusion, and KdV type equations with small dispersion.

We present our recent work on designing third and fourth order accurate weighted essentially non-oscillatory (WENO) schemes for solving the nonlinear Hamilton-Jacobi equations on two-dimensional unstructured meshes. The main ideas are nodal based approximations, the usage of monotone Hamiltonians as building blocks on unstructured meshes, nonlinear weights using smooth indicators of second and higher derivatives, and a strategy to choose diversified smaller stencils to make up the bigger stencil in the WENO procedure. Both third-order and fourth-order WENO schemes use combinations of second-order approximations with nonlinear weights. A brief introduction to the methods and a selected few numerical experiments are included here.

In this paper, we present an overview of the evolution of the discontinuous Galerkin methods since their introduction in 1973 by Reed and Hill, in the framework of neutron transport, until their most recent developments. We show how these methods made their way into the main stream of computational fluid dynamics and how they are quickly finding use in a wide variety of applications. We review the theoretical and algorithmic aspects of these methods as well as their applications to equations including nonlinear conservation laws, the compressible Navier-Stokes equations, and Hamilton-Jacobi-like equations.

The RKDG method has been effectively used in modeling and simulating semiconductor devices, where the underlying models are hydrodynamic in nature. These include classical as well as quantum models. In this paper, we survey and interpret some of these results. For classical transport, we review the simulation of a benchmark MESFET transistor by means of discontinuous Galerkin methods of degree one. For quantum transport, we report the success in simulation of the resonant tunneling diode. The principal features here are negative differential resistance and hysteresis.

In this presentation, we perform further investigation on the least square procedure used in the discontinuous Galerkin methods developed in [2] and [3] for the two-dimensional Hamilton-Jacobi equations. The focus of this presentation will be upon the influence of this least square procedure to the accuracy and stability of the numerical results. We will show through numerical examples that the procedure is crucial for the success of the discontinuous Galerkin methods developed in [2] and [3], especially for high order methods. New test cases using P 4 polynomials, which are at least fourth order and often fifth order accurate, are shown, in addition to the P 2 and P 3 cases presented in [2] and [3]. This addition is non-trivial as the least square procedure plays a more significant role for the P 4 case.

In this presentation we explore a recently introduced high order discontinuous Galerkin method for two dimensional incompressible flow in vorticity streamfunction formulation [8]. In this method, the momentum equation is treated explicitly, utilizing the efficiency of the discontinuous Galerkin method. The streamfunction is obtained by a standard Poisson solver using continuous finite elements. There is a natural matching between these two finite element spaces, since the normal component of the velocity field is continuous across element boundaries. This allows for a correct upwinding gluing in the discontinuous Galerkin framework, while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability. The method is suitable for inviscid or high Reynolds number flows. In our previous work, optimal error estimates are proven and verified by numerical experiments. In this presentation we present one numerical example, the shear layer problem, in detail and from different angles to illustrate the resolution performance of the method.

In these lectures we present the basic ideas and recent development in the construction, analysis, and implementation of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes and their applications to computational fluid dynamics. ENO and WENO schemes are high order accurate finite difference or finite volume schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in computational fluid dynamics and other applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics.

In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics.

In this paper, we present an essentially nonoscillatory spectral Fourier method for the solution of hyperbolic partial differential equations. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that essentially no oscillations develop in the solution. Also, the accuracy of the spectral solution of the inviscid Burgers equation is shown to be higher than a fixed order.