Summation-by-parts operators for correction procedure via reconstruction

The correction procedure via reconstruction (CPR, formerly known as flux reconstruction) is a framework of high order methods for conservation laws, unifying some discontinuous Galerkin, spectral difference and spectral volume methods. Linearly stable schemes were presented by Vincent et al. (2011, 2015), but proofs of non-linear (entropy) stability in this framework have not been published yet (to the knowledge of the authors). We reformulate CPR methods using summation-by-parts (SBP) operators with simultaneous approximation terms (SATs), a framework popular for finite difference methods, extending the results obtained by Gassner (2013) for a special discontinuous Galerkin spectral element method. This reformulation leads to proofs of conservation and stability in discrete norms associated with the method, recovering the linearly stable CPR schemes of Vincent et al. (2011, 2015). Additionally, extending the skew-symmetric formulation of conservation laws by additional correction terms, entropy stability for Burgers' equation is proved for general SBP CPR methods not including boundary nodes.

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

[2]  Gregor J. Gassner,et al.  A kinetic energy preserving nodal discontinuous Galerkin spectral element method , 2014 .

[3]  David C. Del Rey Fernández,et al.  Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations , 2014 .

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

[5]  Magnus Svärd,et al.  Review of summation-by-parts schemes for initial-boundary-value problems , 2013, J. Comput. Phys..

[6]  Freddie D. Witherden,et al.  On the identification of symmetric quadrature rules for finite element methods , 2014, Comput. Math. Appl..

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

[8]  Gregor Gassner,et al.  A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations , 2016, Appl. Math. Comput..

[9]  Antony Jameson,et al.  Connections between the filtered discontinuous Galerkin method and the flux reconstruction approach to high order discretizations , 2011 .

[10]  Antony Jameson,et al.  A New Class of High-Order Energy Stable Flux Reconstruction Schemes for Triangular Elements , 2012, J. Sci. Comput..

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

[12]  Antony Jameson,et al.  Energy stable flux reconstruction schemes for advection-diffusion problems on triangles , 2013, J. Comput. Phys..

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

[14]  Gregor Gassner,et al.  An Energy Stable Discontinuous Galerkin Spectral Element Discretization for Variable Coefficient Advection Problems , 2014, SIAM J. Sci. Comput..

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

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

[17]  H. T. Huynh,et al.  High-Order Methods for Computational Fluid Dynamics: A Brief Review of Compact Differential Formulations on Unstructured Grids , 2013 .

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

[19]  Nail K. Yamaleev,et al.  Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: Theory and boundary conditions , 2013, J. Comput. Phys..

[20]  Jason E. Hicken,et al.  Summation-by-parts operators and high-order quadrature , 2011, J. Comput. Appl. Math..

[21]  Spencer J. Sherwin,et al.  Connections between the discontinuous Galerkin method and high‐order flux reconstruction schemes , 2014 .

[22]  Gregor Gassner,et al.  A Comparison of the Dispersion and Dissipation Errors of Gauss and Gauss-Lobatto Discontinuous Galerkin Spectral Element Methods , 2011, SIAM J. Sci. Comput..

[23]  Zhi Jian Wang,et al.  On the Connection Between the Correction and Weighting Functions in the Correction Procedure via Reconstruction Method , 2013, J. Sci. Comput..

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

[25]  Gregor Gassner,et al.  A Skew-Symmetric Discontinuous Galerkin Spectral Element Discretization and Its Relation to SBP-SAT Finite Difference Methods , 2013, SIAM J. Sci. Comput..

[26]  David C. Del Rey Fernández,et al.  A generalized framework for nodal first derivative summation-by-parts operators , 2014, J. Comput. Phys..

[27]  Zhi Jian Wang,et al.  A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids , 2009, J. Comput. Phys..

[28]  Gregor Gassner,et al.  On the Quadrature and Weak Form Choices in Collocation Type Discontinuous Galerkin Spectral Element Methods , 2010, J. Sci. Comput..