# Prof. Georg May Ph.D.

### Faculty

High-Order Methods in Multiphase Flow Simulation

Mechanical Engineering

### Member

Junior Research Group Leader

### Research

High Order Schemes for PDE, Aerodynamics

## Contact

Schinkelstr. 2

52056 Aachen

Germany

Phone: +49-241-8099133

Fax: +49-241-80628498

email: may@aices.rwth-aachen.de

## Education

- B.E. Thayer School of Engineering, Dartmouth College, Hanover, NH, 2000
- Dipl. Ing., Institut für Luft- und Raumfahrt, RWTH Aachen, Germany, 2001
- PhD, Departmenf of Aeronautics and Astronautics, Stanford University, Stanford, CA, 2006

## Professional Career

- 2006-2007 - Research Associate, Departmenf of Aeronautics and Astronautics, Stanford University, Stanford, CA
- 2007-2009 - Junior Research Group Leader, Graduate School AICES, RWTH Aachen, Aachen, Germany
- 2009-2014 - Junior Professor (W1), Graduate School AICES, RWTH Aachen, Aachen, Germany
- since 2014 - Associate Professor (W2), Graduate School AICES, RWTH Aachen, Aachen, Germany

## 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

- Dolejsi, V., May, G., Rangarajan, A.: "
*A continuous hp-mesh model for adaptive discontinuous Galerkin schemes*", Appl. Numer. Math., in Press. - Zakerzaedeh, M,. May, G."
*Class of Space-Time Entropy Stable DG Schemes for Systems of Convection-Diffusion*", Hyp2016, in Springer Proceedings in Mathematics and Statistics, to appear. - Dolejsi, V., May, G., Roskovec, F., Solin, P.: "
*Anisotropic hp-mesh optimization technique based on the continuous mesh and error models*", Comput. Math. Appl. 74(1), pp. 45-63, 2017. - Rangarajan, A., May, G., Dolejsi, V.: "Adjoint-based anisotropic mesh adaptation for Discontinuous Galerkin Methods Using a Continuous Mesh Model", AIAA Paper 17-3100, American Institute of Aeronautics and Astronautics, 2017.
- Zakerzadeh, M., May, G. "
*Entropy stable Discontinuous Galerkin Scheme for the compressible Navier-Stokes Equations*", AIAA Paper 17-0084, American Institute of Aeronautics and Astronautics, 2017. - May, G., Zakerzadeh, M.:
*"On the Convergence of Space-Time Discontinuous Galerkin Schemes for Scalar Conservation Laws"*, SIAM J. Numer. Anal. 54(4), pp. 2452-2465, 2016. - Zakerzadeh, M., May, G.: "
*High-Order Entropy Stable Discontinuous Galerkin Schemes in a Space-Time Computational Framework*", AIAA Paper 16-3496, American Institute of Aeronautics and Astronautics, 2016. - Balan, A., Woopen, M., May, G.:
*"Adjoint-based hp-adaptivity on anisotropic meshes for high-order compressible flow simulations"*, Computers & Fluids 139, pp. 47-67, 2016. - Rangarajan, A., Balan, A., May, G.:
*"Mesh Adaptation and Optimization for Discontinuous Galerkin Methods Using a Continuous Mesh Model"*, AIAA Paper 16-2142, American Institute of Aeronautics and Astronautics, 2016. - Zakerzadeh, M., May, G.:
*"On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws"*, SIAM J. Numer. Anal. 54(2), pp. 874-898 , 2016 - May, G.:
*"Hybridized Discontinuous Galerkin Methods: Formulation and Discrete Adjoint"*, in VKI Lecture Series 2015-04: 38th Advanced Computational Fluid Dynamics, Deconinck, H., Horvath, T. (ed.), von Karman Institute for Fluid Dynamics, 2015. - May, G., Balan, A., Woopen, M.:
*"Output-based error estimation and mesh adaptation for HDG methods"*, in VKI Lecture Series 2015-04: 38th Advanced Computational Fluid Dynamics, Deconinck, H., Horvath, T. (ed.), von Karman Institute for Fluid Dynamics, 2015. - Woopen, M., Balan, A., May, G.:
*"A Unifying Computational Framework for Adaptive High-Order Finite Element Methods"*, AIAA Paper 15-2601, American Institute of Aeronautics and Astronautics, 2015. - Woopen, M., May, G.:
*"An Anisotropic Adjoint-Based hp-Adaptive HDG Method for Compressible Turbulent Flow"*, AIAA Paper 15-2042, American Institute of Aeronautics and Astronautics, 2015. - 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. 62 (1), pp. 25-52, 2015. - Woopen, M, May, G., and Schütz, J.:
*"Adjoint-based error estimation and mesh adaptation for hybridized discontinuous Galerkin methods"*, Int. J. Numer. Meth. Fluids 76(11), pp. 811–834, 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. - 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. 34 (3), pp. 1222-1239, 2014. - 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., Fidkowski, K. et al. (May, G.):
*"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"*, 40^{th}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"*, 19^{th}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"*, 47^{th}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, 44^{th}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, 17^{th}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, 17^{th}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, 43^{rd}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, 43^{rd}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, 42^{nd}AIAA Aerospace Sciences Meeting & Exhibit, Reno, NV, January 5-8, 2004.