High Order Discontinuous Galerkin Methods for Aerodynamics
by
Jaime Peraire
Department of Aeronautics and Astronautics, Massachusetts Institute of Technology
Coauthors: P.-O. Persson

Discontinuous Galerkin (DG)[1] methods for solving the Navier-Stokes equations have received considerable attention in recent years because of their potential to produce highly accurate solutions with minimum numerical dissipation. Applications such as LES or aero-acoustics demand high accuracy and low dispersion and hence are clear candidates for high order methods. DG methods produce stable discretizations of the convective operator for arbitrary accuracy orders. Moreover, Discontinuous Galerkin methods can be used with unstructured meshes of tetrahedra, which appears to be a requirement for real-world complex geometries.

Until now, however, DG methods have only been demonstrated in the research community for academic applications with relatively simple flows and geometries. One of the main challenges is to extend these methods to relevant engineering applications. This requires the ability to generate appropriately stretched higher order meshes, perform stable discretizations for the for the Navier-Stokes equations, maintain stability int he presence of shocks and solve the resulting systems of equations in a competitive manner.

We will present our work on Discontinuous Galerkin methods for aerodynamic applications. In particular, will describe our low cost discretization of the viscous terms [2], efficeint solution methods [3], our efforts in shock capturing [4] as well as the extension of the method to handle RANS flows [5]. We will also describe several applications involving fluid structure interaction [6] with particular emphasis on flapping flight.

REFERENCES

[1] Bernardo Cockburn and C.-W. Shu, ”Runge-Kutta Discontinuous Galerkin Methods for Covective-Dominated Problems”, Review

Article, J. Sci. Comp., v. 16, pp.173-261, 2001.

[2] Peraire, J., and Persson, P.-O., The compact discontinuous Galerkin (CDG) method for elliptic problems, to appear SIAM Journal

for Scientific Computing, 2006.

[3] Persson, P.-O., and Peraire, J., An Efficient Low Memory Implicit Discontinuous Galerkin Algorithm for Time Dependent Problems,

AIAA-2006-0113, Reno, January 2006.

[4] P.-O. Persson and J. Peraire, ”Sub-cell shock capturing for discontinuous Galerkin methods”, AIAA-2006-0112, 44th Aerospace

Sciences Meeting, Reno, Nevada, 2006.

[5] Nguyen, N.C., Persson, P.-O., Peraire, J., Bonet, J., ”RANS Solutions using high order discontinuous Galerkin methods,” AIAA

Aerospace Sciences Meeting & Exhibit, AIAA-2007-0914, AIAA, Reno, NV, 2007.

[6] Persson, P.-O., Peraire, J., Bonet, J., ”Discontinuous Galerkin Solution of the Navier-Stokes Equations on Deformable Domains,”

AIAA Aerospace Sciences Meeting & Exhibit, AIAA-2007-513, AIAA, Reno, NV, 2007.

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Date received: September 3, 2007