Geometry effects in nodal discontinuous Galerkin methods on curved elements that are provably stable

We investigate three effects of the variable geometric terms that arise when approximating linear conservation laws on curved elements with a provably stable skew-symmetric variant of the discontinuous Galerkin spectral element method (DGSEM). We show for a constant coefficient system that the non-constant coefficient problem generated by mapping a curved element to the reference element is stable and has energy bounded by the initial value as long as the discrete metric identities are satisfied. Under those same conditions, the skew-symmetric approximation is also constant state preserving and discretely conservative, just like the original DGSEM.

[1]  Jeremy E. Kozdon,et al.  Simulation of Dynamic Earthquake Ruptures in Complex Geometries Using High-Order Finite Difference Methods , 2013, J. Sci. Comput..

[2]  Marc Duruflé,et al.  Application of Discontinuous Galerkin spectral method on hexahedral elements for aeroacoustic , 2009 .

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

[4]  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..

[5]  W. J. Gordon,et al.  Construction of curvilinear co-ordinate systems and applications to mesh generation , 1973 .

[6]  Joel Ferziger,et al.  Higher Order Methods for Incompressible Fluid Flow: by Deville, Fischer and Mund, Cambridge University Press, 499 pp. , 2003 .

[7]  M. Y. Hussaini,et al.  Discontinuous Spectral Element Solution of Acoustic Radiation from Thin Airfoils , 2001 .

[8]  Claus-Dieter Munz,et al.  Explicit Discontinuous Galerkin methods for unsteady problems , 2012 .

[9]  David A. Kopriva,et al.  Metric Identities and the Discontinuous Spectral Element Method on Curvilinear Meshes , 2006, J. Sci. Comput..

[10]  M. Y. Hussaini,et al.  Discontinuous Spectral Element Approximation of Maxwell’s Equations , 2000 .

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

[12]  F. Farassat,et al.  Aircraft Engine Noise Scattering - A Discontinuous Spectral Element Approach , 2002 .

[13]  Davis A. Kopriva,et al.  Computation of electromagnetic scattering with a non‐conforming discontinuous spectral element method , 2002 .

[14]  M. Y. Hussaini,et al.  Computation of Engine Noise Propagation and Scattering off An Aircraft , 2002 .

[15]  Shaozhong Deng,et al.  Numerical simulation of optical coupling and light propagation in coupled optical resonators with size disorder , 2007 .

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

[17]  Wei Cai,et al.  Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides. , 2004, Optics express.

[18]  Claus-Dieter Munz,et al.  An Efficient High Performance Parallelization of a Discontinuous Galerkin Spectral Element Method , 2012, Facing the Multicore-Challenge.

[19]  Anthony T. Patera,et al.  An isoparametric spectral element method for solution of the Navier-Stokes equations in complex geometry , 1986 .

[20]  Jan Nordström,et al.  Conservative Finite Difference Formulations, Variable Coefficients, Energy Estimates and Artificial Dissipation , 2006, J. Sci. Comput..

[21]  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..

[22]  Gary Cohen,et al.  A spatial high-order hexahedral discontinuous Galerkin method to solve Maxwell's equations in time domain , 2006, J. Comput. Phys..

[23]  A. Quarteroni,et al.  Approximation results for orthogonal polynomials in Sobolev spaces , 1982 .