Temporal Stabilisation of Flux Reconstruction on Linear Problems

Filtering is often used in Large Eddy Simulation with a global filter width, instead here a filter width in the reference domain of high order Flux Reconstruction is considered. It is shown via Von Neumann analysis how filtering effects the dispersion and dissipation of the scheme when spatially and temporally discretised. With it being shown that filtering stabilises the scheme temporally by upto $25\%$ for forth order FR. The impact of filtering on error production is calculated, highlighting the reduction in convective velocity caused and showing numerically the impact on order of accuracy. Finally, the turbulent Taylor-Green case is used to understand the effect of reference domain filtering on the transition to turbulence, and a filter Reynolds number is defined that is shown to be useful in understanding the effect of filtering on simulations.

[1]  S. Ghosal.  An Analysis of Numerical Errors in Large-Eddy Simulations of Turbulence , 1996 .

[2]  Antony Jameson,et al.  On consistency and rate of convergence of Flux Reconstruction for time-dependent problems , 2017, J. Comput. Phys..

[3]  Brian C. Vermeire,et al.  On the behaviour of fully-discrete flux reconstruction schemes , 2017 .

[4]  D. Carati,et al.  Large-eddy simulation , 2000 .

[5]  Jonathan R. Bull,et al.  Simulation of the Compressible Taylor Green Vortex using High-Order Flux Reconstruction Schemes , 2014 .

[6]  H. T. Huynh,et al.  A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods , 2007 .

[7]  A. Jameson,et al.  High-Order Flux Reconstruction Schemes , 2016 .

[8]  Freddie D. Witherden,et al.  PyFR: An open source framework for solving advection-diffusion type problems on streaming architectures using the flux reconstruction approach , 2013, Comput. Phys. Commun..

[9]  S. Orszag,et al.  Small-scale structure of the Taylor–Green vortex , 1983, Journal of Fluid Mechanics.

[10]  J. Debonis.  Solutions of the Taylor-Green Vortex Problem Using High-Resolution Explicit Finite Difference Methods , 2013 .

[11]  Freddie D. Witherden,et al.  An extended range of stable-symmetric-conservative Flux Reconstruction correction functions , 2015 .

[12]  F. D. Witherden,et al.  An Analysis of Solution Point Coordinates for Flux Reconstruction Schemes on Tetrahedral Elements , 2016, J. Sci. Comput..

[13]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[14]  Antony Jameson,et al.  Non-linear stabilization of high-order Flux Reconstruction schemes via Fourier-spectral filtering , 2015, J. Comput. Phys..

[15]  Will Trojak,et al.  High-Order Flux Reconstruction on Stretched and Warped Meshes , 2017, AIAA Journal.

[16]  Antony Jameson,et al.  On the Non-linear Stability of Flux Reconstruction Schemes , 2012, J. Sci. Comput..

[17]  Peretz P. Friedmann,et al.  Announcements, Comments, and Acknowledgments , 2011 .

[18]  Freddie D. Witherden,et al.  Towards Green Aviation with Python at Petascale , 2016, SC16: International Conference for High Performance Computing, Networking, Storage and Analysis.

[19]  S. Lele.  Compact finite difference schemes with spectral-like resolution , 1992 .

[20]  David A. Pope.  An exponential method of numerical integration of ordinary differential equations , 1963, CACM.

[21]  Antony Jameson,et al.  A New Class of High-Order Energy Stable Flux Reconstruction Schemes , 2011, J. Sci. Comput..

[22]  Will Trojak,et al.  Generalised Sobolev Stable Flux Reconstruction , 2018, 1804.04714.

[23]  Antony Jameson,et al.  Insights from von Neumann analysis of high-order flux reconstruction schemes , 2011, J. Comput. Phys..

[24]  A. Jameson.  High-Order Methods for Diffusion Equation with Energy Stable Flux Reconstruction Scheme , 2011 .

[25]  Antony Jameson,et al.  High-Order Flux Reconstruction Schemes with Minimal Dispersion and Dissipation , 2015, J. Sci. Comput..

[26]  Rolf Rannacher,et al.  Time-Dependent Problems , 2003 .

[27]  Antony Jameson,et al.  Energy Stable Flux Reconstruction Schemes for Advection–Diffusion Problems on Tetrahedra , 2014, J. Sci. Comput..

[28]  Antony Jameson,et al.  Energy Stable Flux Reconstruction Schemes for Advection–Diffusion Problems on Tetrahedra , 2013, Journal of Scientific Computing.

[29]  Freddie D. Witherden,et al.  An Analysis of Solution Point Coordinates for Flux Reconstruction Schemes on Triangular Elements , 2014, J. Sci. Comput..

[30]  Julia S. Mullen,et al.  Filter-based stabilization of spectral element methods , 2001 .