Application of High-Order Energy Stable Flux Reconstruction Schemes to the Euler Equations

The authors recently identified an infinite range of high-order energy stable flux reconstruction (FR) schemes in 1D and on triangular elements in 2D. The new flux reconstruction schemes are linearly stable for all orders of accuracy in a norm of Sobolev type. They are parameterized by a single scalar quantity, which if chosen judiciously leads to the recovery of various well known high-order methods (such as a collocation based nodal discontinuous Galerkin method and a spectral difference method). Identification of such schemes represents a significant advance in terms of understanding why certain FR schemes are stable, whereas others are not. However, to date there have been no studies into how these schemes perform when applied to real world non-linear problems. In this paper, stability and accuracy properties of these new schemes are studied for various two-dimensional inviscid flow problems. The results offer significant insight into the performance of energy stable FR schemes for non-linear problems. It is envisaged the results will aid scheme selection for a given problem, based on its stability and accuracy requirements.

[1]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[2]  Chunlei Liang,et al.  A p-multigrid spectral difference method with explicit and implicit smoothers on unstructured triangular grids , 2009 .

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

[4]  Marcel Vinokur,et al.  Spectral difference method for unstructured grids I: Basic formulation , 2006, J. Comput. Phys..

[5]  M. Carpenter,et al.  Fourth-order 2N-storage Runge-Kutta schemes , 1994 .

[6]  Antony Jameson,et al.  Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations , 2007, J. Sci. Comput..

[7]  Antony Jameson,et al.  A Proof of the Stability of the Spectral Difference Method for All Orders of Accuracy , 2010, J. Sci. Comput..

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

[9]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

[10]  Beth A. Wingate,et al.  Several new quadrature formulas for polynomial integration in the triangle , 2005 .

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

[12]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[13]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[14]  Chunlei Liang,et al.  Simulation of Transitional Flow over Airfoils using the Spectral Difference Method , 2010 .

[15]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[16]  Chi-Wang Shu.  Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws , 1998 .

[17]  John H. Kolias,et al.  A CONSERVATIVE STAGGERED-GRID CHEBYSHEV MULTIDOMAIN METHOD FOR COMPRESSIBLE FLOWS , 1995 .