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The development of high-order accurate numerical discretization techniques for irregular domains and meshes is often cited as one of the remaining chal­ lenges facing the field of computational fluid dynamics. In structural me­ chanics, the advantages of high-order finite element approximation are widely recognized. This is especially true when high-order element approximation is combined with element refinement (h-p refinement). In computational fluid dynamics, high-order discretization methods are infrequently used in the com­ putation of compressible fluid flow. The hyperbolic nature of the governing equations and the presence of solution discontinuities makes high-order ac­ curacy difficult to achieve. Consequently, second-order accurate methods are still predominately used in industrial applications even though evidence sug­ gests that high-order methods may offer a way to significantly improve the resolution and accuracy for these calculations. To address this important topic, a special course was jointly organized by the Applied Vehicle Technology Panel of NATO's Research and Technology Organization (RTO), the von Karman Institute for Fluid Dynamics, and the Numerical Aerospace Simulation Division at the NASA Ames Research Cen­ ter. The NATO RTO sponsored course entitled "Higher Order Discretization Methods in Computational Fluid Dynamics" was held September 14-18,1998 at the von Karman Institute for Fluid Dynamics in Belgium and September 21-25,1998 at the NASA Ames Research Center in the United States.



High Order Approximations for Compressible Fluid Dynamics on Unstructured and Cartesian Meshes

We describe in detail some techniques to construct high order MUSCL type schemes on general meshes : the ENO and WENO type schemes. Special attention is given to the reconstruction step. Extesio to Hamilton Jacobi equations is sketched. We also present some hybrid techniques that use simple modifications of classical TVD schemes yielding in a very clear improvements of the accuracy. We discuss means of improving the efficiency using Harten’s multiresolution analysis. We provide several numerical examples and comparisions with more conventional schemes.
Rémi Abgrall, Thomas Sonar, Oliver Friedrich, Germain Billet

Discontinuous Galerkin Methods for Convection-Dominated Problems

We present and analyze the Runge Kutta Discontinuous Galerkin method for numerically solving nonlinear hyperbolic systems. The basic method is then extended to convection-dominated problems yielding the Local Discontinuous Galerkin method. These methods are particularly attractive since they achieve formal high-order 0accuracy, nonlinear stability, and high parallelizability while maintaining the ability to handle complicated geometries and capture the discontinuities or strong gradients of the exact solution without producing spurious oscillations. The discussed methods are readily applied to the Euler equations of gas dynamics, the shallow water equations, the equations of magneto-hydrodynamics, the compressible Navier-Stokes equations with high Reynolds numbers, and the equations of the hydrodynamic model for semiconductor device simulation. As a final example, consideration is given to the application of the discontinuous Galerkin method to the Hamilton-Jacobi equations.
Bernardo Cockburn

Adaptive Spectral Element Methods for Turbulence and Transition

These notes present an introduction to the spectral element method with applications to fluid dynamics. The method is introduced for one-dimensional problems, followed by the discretization of the advection and diffusion operators in multi-dimensions, and efficient ways of dealing with these operators numerically. We also discuss the mortar element method, a technique for incorporating local mesh refinement using nonconforming elements; this is the foundation for adaptive methods. An adaptive strategy based on analyzing the local polynomial spectrum is presented and shown to give accurate solutions even for problems with weak singularities. Finally we describe techniques for integrating the incompressible Navier-Stokes equations, including methods for performing computational linear and nonlinear stability analysis of non-parallel and time-periodic flows.
Ronald D. Henderson

hp-FEM for Fluid Flow Simulation

We present some mathematical foundations of hp-FEM for fluid flow simulation. Particular attention is paid to the mesh-design for viscous, incompressible flow where the regularity of the solution mandates resolution of corner singularities and boundary layers. Stabilized and discontinuous hp-FEM for advection dominated and nearly incompressible flows are derived. A new hp-adaptive time stepping strategy for spectral accuracy in transient problems is presented.
Christoph Schwab

High Order ENO and WENO Schemes for Computational Fluid Dynamics

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.
Chi-Wang Shu


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