High order accurate weighted essentially nonoscillatory (WENO) schemes are relatively new but have gained rapid popularity in numerical solutions of hyperbolic partial differential equations (PDEs) 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, nonoscillatory, 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 user to tune parameters. At the heart of the WENO schemes is actually an approximation procedure not directly related to PDEs, hence the WENO procedure can also be used in many non-PDE applications. 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 PDEs and other convection dominated problems, and present a collection of applications in areas including computational fluid dynamics, computational astronomy and astrophysics, semiconductor device simulation, traffic flow models, computational biology, and some non-PDE applications. Finally, we mention a few topics concerning WENO schemes that are currently under investigation.

High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems

Fixed membrane wings for micro air vehicles: Experimental characterization, numerical modeling, and tailoring

https://www.brown.edu/research/projects/scientific-computing/sites/brown.edu.research.projects.scientific-computing/files/uploads/High%20order%20WENO%20and%20DG%20methods%20for%20time-dependent%20convection-dominated%20PDEs.pdf

High order WENO and DG methods for time-dependent convection-dominated PDEs

Development of nonlinear weighted compact schemes with increasingly higher order accuracy

The unsteady flow past a NACA 0012 airfoil that is undertaking a constant-rate pitching up motion is investigated experimentally by the PIDV technique in a water towing tank. The Reynolds number is 5000, based upon the airfoil's chord and the free-stream velocity. The airfoil is pitching impulsively from 0 to 30 deg. with a dimensionless pitch rate alpha of 0.131. Instantaneous velocity and associated vorticity data have been acquired over the entire flow field. The primary vortex dominates the flow behavior after it separates from the leading edge of the airfoil. Complete stall emerges after this vortex detaches from the airfoil and triggers the shedding of a counter-rotating vortex near the trailing edge. A parallel computational study using the discrete vortex, random walk approximation has also been conducted. In general, the computational results agree very well with the experiment.

Unsteady flow past an airfoil pitching at a constant rate

Direct numerical simulation of flow separation around a NACA 0012 airfoil

Numerical study of passive and active flow separation control over a NACA0012 airfoil

In this paper a new class of finite difference schemes - the Weighted Compact Schemes are proposed. According to the idea of the WENO schemes, the Weighted Compact Scheme is constructed by a combination of the approximations of derivatives on candidate stencils with properly assigned weights so that the non-oscillatory property is achieve when discontinuities appear. The primitive function reconstruction method of ENO schemes is applied to obtain the conservative form of the Weighted Compact Scheme. This new scheme not only preserves the characteristic of standard compact schemes and achieves high order accuracy and high resolution using a compact stencil, but also can accurately capture shock waves and discontinuities without oscillation. Numerical examples show the new scheme is very promising and successful.

Weighted Compact Scheme for Shock Capturing

: In this paper a new class of finite difference schemes - the Weighted Compact Schemes are proposed According to the idea of the WENO schemes, the Weighted Compact Scheme is constructed by a combination of the approximations of derivatives on candidate stencils with properly assigned weights so that the non-oscillatory property is achieved when discontinuities appear The primitive function reconstruction method of ENO schemes is applied to obtain the conservative form of the Weighted Compact Scheme This new scheme not only preserves the characteristic of standard compact schemes and achieves high order accuracy and high resolution using a compact stencil, but also can accurately capture shock waves and discontinuities without oscillation Numerical examples show the new scheme is very promising and successfill

Weighted Compact Scheme

The immersed boundary method with discrete direct forcing approach was combined with a direct numerical simulation (DNS) code to study the time-dependent response of boundary-layer flow over a flat plate to the active vortex generators (AVG) that consist of a pair of deployable circular wing lip type blades driven by a pre-defined duty cycle. The DNS code solves the three-dimensional Navier-Stokes equations for compressible flow in general curvilinear coordinates using a fully implicit LU-SGS method. A fourth-order finite difference scheme is used to compute the spatial derivatives. The underlying curvilinear mesh near the vortex generator is designed properly with a large portion of the immersed boundary intersecting with the grid nodes to reduce the number of near-boundary nodes that require interpolation. The current immersed boundary technique was tested and verified on backward-facing step flow and flow past a circular cylinder. Numerical experiments have shown that the discrete direct forcing approach is very effective. In numerical simulation of flow behind the AVG, the computational results are comparable to the experimental data.

Hua Shan

Papers

Direct numerical simulation of flow separation around a NACA 0012 airfoil

Numerical study of passive and active flow separation control over a NACA0012 airfoil

Weighted Compact Scheme for Shock Capturing

Weighted Compact Scheme

Numerical simulation of flow behind active vortex generators with direct forcing immersed boundary method