Skip to content. | Skip to navigation


Sections
Document Actions

Prof. Georg May

Contact


Schinkelstr. 2
52056 Aachen
Germany
Phone: +49-241-8099133
Fax: +49-241-80628498
email: see here...

Education



Professional Career



Research Interests


High-Order Numerical Methods for Conservation Laws

Here the focus is on high-order accurate numerical methods for partial differential equations (PDE), in particular those of the hyperbolic type. I am interested in suitable high-order accurate algorithms for general unstructured meshes.

There is considerable demand for the numerical solution of differential equations in rather complex computational domains. For example, the Euler or Navier-Stokes equations are routinely solved for full aircraft configuration by lower order methods. This has motivated the use of unstructured mesh technologies, which makes meshing of the computational domain simple, but the formulation of high-order accurate numerical algorithms much more challenging.

Hyperbolic PDE are known to admit discontinuous solutions in finite time, even for smooth initial data. A well known example are the compressible Euler equations, which allow compression shocks for sufficiently high Mach numbers. This necessitates the development of shock capturing schemes, which have matured over the years for ower order accuracy, and are widely used in industrial applications. Extensions to high order schemes, however, still leave a lot to be desired.

I have been involved in the development of the Spectral Difference (SD) Method. The SD method uses a local pseudo-spectral representation of the solution on unstructured meshes to achieve arbitrary order of accuracy. Liu, Vinokur and Wang proposed the baseline scheme for hyperbolic PDE in 2004. My own research, which has been carried out at the Aerospace Computing Laboratory at Stanford University in collaboration with Professor ZJ Wang at Iowa State University, has concentrated on the following contributions:

  • Investigation of linear and nonlinear stability properties
  • Extension of the scheme to the compressible Navier-Stokes equations
  • Ensure stability and monotonicity in transonic and supersonic fluid flow through appropriate limiting procedures
  • Development of an h/p-Multigrid procedure

Kinetic Schemes for the Navier-Stokes Equations

Conventional techniques for the Navier-Stokes equations necessitate the use of different discretization stecils for the inviscid and viscous flux components. While these techniques often have a physically sound motivation, algorithmically it can be quite challenging to discretize the Navier-Stokes equations on general meshes, i.e. arbitrary polyhedra.

Gas-Kinetic finite volume schemes offer an attractive alternative in that they allow the discretization of the Navier-Stokes equations on a simple and universal next-neighbor stencil. This is accomplished by applying the discretization to the gas-kinetic distribution function, rather than the macroscopic variables. The connection between the mesoscopic and macroscopic level is established by the governing gas-kinetic equations, i.e. the Boltzmann or BGK equations and related framework from kinetic gas theory, such as the Chapman-Enskog expansion.

The discretization of the Navier Stokes equations can be accomplished by first computing gradients of the macroscopic variables, and subsequent discretization of a reconstructed gas-kinetic distribution function, which is a function of the macroscopic variables and their gradients, and must be consistent to the Navier-Stokes order of accuracy in Chapman-Enskog expansion. Written in terms of the macroscopic variables this amounts to a nonlinear reconstruction and averaging of flow variables and their gradients to approximate convective and viscous fluxes.

Numerical Algorithms for General Unstructured Meshes

I am interested in the software engineering aspects related to creating flexible computational tools for modeling physics on general unstructured meshes. There are numerous challenges related to computational geometry, algorithmic details (such as multigrid techniques) for solvers that operate on arbitrary polyhedral meshes. This is true even for low-order approximations, and for any type of physics problem. In this context I have been involved in the design and developed of computational architectures for modeling compressible fluid flow on arbitrary polyhedral meshes.

Publications


Woopen, M, May, G., and Schütz, J.: "Adjoint-based error estimation and mesh adaptation for hybridized discontinuous Galerkin methods", Int. J. Numer. Meth. Fluids, in press, 2014.

Woopen, M, Balan, A., May, G., and Schütz, J.: "A Comparison of Hybridized and Standard DG Methods for Target-Based hp-Adaptive Simulation of Compressible Flow", Computers & Fluids 98, pp. 3–16, 2014.

Gerhard, N., Iacono, F., May, G., Müller, S., Schäfer, R.: "A High-Order Discontinuous Galerkin Discretization with Multiwavelet-Based Grid Adaptation for Compressible Flows", J. Sci. Comput. in press, 2014.

Balan, A. Woopen, M., May, G.: "A Hybridized Discontinuous Galerkin Method for Three-Dimensional Compressible Flow Problems", AIAA Paper 14-0938, American Institute of Aeronautics and Astronautics, 2014.

Schütz, J., May, G.: "An Adjoint Consistency Analysis for a Class of Hybrid Mixed Methods", IMA J. Numer. Anal. in press, 2013.

Balan, A. Woopen, M., May, G.: "Adjoint-Based Hp-Adaptation for a Class of High-Order Hybridized Finite Element Schemes for Compressible Flows", AIAA Paper 13-2938, American Institute of Aeronautics and Astronautics, 2013.

J. Schütz, M. Woopen and G. May: "A combined hybridized discontinuous Galerkin / hybrid mixed method for viscous conservation laws", in: Hyperbolic Problems: Theory, Numerics, Applications, Proceedings of the Fourteenth International Conference on Hyperbolic Problems, Padova, Italy, pp. 915-922, AIMS, 2013.

Wang, Z. J., May, G. et al.: "High-order CFD methods: current status and perspective", Int. J. Numer. Meth. Fluids 72(8), pp. 811–845, 2013.

Schütz J., May, G.: "A Hybrid Mixed Method for the Compressible Navier-Stokes Equations", J. Comp. Phys 240, pp. 58-75, 2013.

J. Schütz, S. Noelle, C. Steiner, May, G.: "A Note on Adjoint Error Estimation for One-Dimensional Stationary Conservation Laws with Shocks", SIAM J. Numerical Analysis 51(1), pp. 126-136, 2013

Schütz, J., Woopen, M., May, G.: "A Hybridized DG/Mixed Scheme for Nonlinear Advection-Diffusion Systems, Including the Compressible Navier-Stokes Equations ", AIAA Paper 12-0729, American Institute of Aeronautics and Astronautics, 2012.

Balan, A., Schöberl, J.,May, G.: "A Stable High-Order Spectral Difference Method for Hyperbolic Conservation Laws on Triangular Elements", J. Comput. Phys. 231(5), pp. 2359–2375, 2012.

May, G., Iacono, F., Balan, A.: "Time-Relaxation Methods for High-Order Discretization of Compressible Flow Problems", Proceedings of the International Conference on Numerical Analysis and Applied Mathematics 2011 (ICNAAM‐2011), AIP Conf. Proc. 1389, 903-906 (2011).

Iacono, F., May, G., Müller, S. and Schäfer, R.: "A Discontinuous Galerkin Discretization with Multiwavelet-Based Grid Adaptation for Compressible Flows", AIAA Paper 11-0200, American Institute of Aeronautics and Astronautics, 2011.

Schütz J., May, G.: "A Numerical Study of Adjoint-Based Mesh Adaptation for Compressible Flow Simulation", AIAA Paper 11-0213, American Institute of Aeronautics and Astronautics, 2011.

May, G.: "On the connection between the spectral difference method and the discontinuous Galerkin method", Commun. Comput. Phys., 9, pp.1071-1080, 2011.

May, G., Jameson, A.: "Efficient Relaxation Methods for High-Order Discretization of Steady Problems", in: Adaptive high-order methods in Computational Fluid Dynamics (Wang, Z. J., ed.), World Scientific Publishing, 2011.

Balan, A., May, G., Schöberl: "A Stable Spectral Difference Method for Triangles", AIAA Paper 11-0047, American Institute of Aeronautics and Astronautics, 2011.

Iacono, F., May, G., Wang, Z. J.: "Relaxation Techniques for High-Order Discretizations of Steady Compressible Inviscid Flows", 40th AIAA Computational Fluid Dynamics Conference, Chicago, IL, AIAA Paper 10-4991, 2010.

G. May, F. Iacono, A. Jameson: " A hybrid multilevel method for high-order discretization of the Euler equations on unstructured meshes", J. Comp. Phys. 229(10), 2010.

Iacono, F., May, G.: "Convergence Acceleration for Simulation of Steady-State Compressible Flows Using High-Order Schemes", 19th AIAA Computational Fluid Dynamics Conference, San Antonio, TX, AIAA Paper 09-4132, 2009.

Iacono, F., May, G., Jameson, A.: "Efficient Algorithms for High-Order Discretization of the Euler and Navier-Stokes Equations", 47th AIAA Aerospace Sciences Meeting and Exhibit, Orlando, FL, AIAA Paper 09-0182, 2009.

May, G., Jameson, A.: "An improved gas-kinetic BGK finite-volume method for three-dimensional transonic flow" ,J. Comp. Phys. 220(2), pp. 856-878, 2007

Wang, Z.J., Liu, Y., May, G., Jameson, A.: "Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations" , J. Sci. Comput. 32 (1) pp. 54-71, July, 2007

May, G., Jameson, A.: "A Spectral Difference Method for the Euler and Navier-Stokes Equations on Unstructured Meshes", AIAA-2006-0304, 44th AIAA Aerospace Sciences Meeting and Exhibit, January 9-12, 2006, Reno, NV.

May, G., Jameson, A.: "High-Order Accurate Methods for High-Speed Flow", AIAA-2005-5251, 17th AIAA Computational Fluid Dynamics Conference, Toronto, Ontario, June 6-9, 2005.

May, G., Jameson, A.: "Improved Gaskinetic Multigrid Method for Three-Dimensional Computation of Viscous Flow", AIAA-2005-5106, 17th AIAA Computational Fluid Dynamics Conference, Toronto, Ontario, June 6-9, 2005.

May, G., Jameson, A.: "Unstructured Algorithms for Inviscid and Viscous Flows Embedded in a Unified Solver Architecture: Flo3xx", AIAA-2005-0318, 43rd AIAA Aerospace Sciences Meeting & Exhibit, Reno, NV, January 10-13, 2005.

May, G., Jameson, A.: "Calculating Three-Dimensional Transonic Flow Using a Gas-Kinetic BGK Finite-Volume Method", AIAA-2005-1397, 43rd AIAA Aerospace Sciences Meeting & Exhibit, Reno, NV, January 10-13, 2005.

May, G., Van der Weide, E., Jameson, A., and Shankaran, S.: "Drag Prediction of the DLR-F6 configuration", AIAA-2004-0396, 42nd AIAA Aerospace Sciences Meeting & Exhibit, Reno, NV, January 5-8, 2004.