Application of modal filtering to a spectral difference method

Glaubitz, Jan; Öffner, Philipp; Sonar, Thomas

doi:10.1090/mcom/3257

Application of modal filtering to a spectral difference method
HTML articles powered by AMS MathViewer

by Jan Glaubitz, Philipp Öffner and Thomas Sonar;

Math. Comp. 87 (2018), 175-207

DOI: https://doi.org/10.1090/mcom/3257

Published electronically: August 7, 2017

PDF | Request permission

Abstract:

We adapt the spectral viscosity (SV) formulation implemented as a modal filter to a Spectral Difference Method (SD) solving hyperbolic conservation laws. In the SD Method we use selections of different orthogonal polynomials (APK polynomials). Furthermore we obtain new error bounds for filtered APK extensions of smooth functions. We demonstrate that the modal filter also depends on the chosen polynomial basis in the SD Method. Spectral filtering stabilizes the scheme and leaves weaker oscillations. Hence, the selection of the family of orthogonal polynomials on triangles and their specific modal filter possesses a positive influence on the stability and accuracy of the SD Method. In the second part, we initiate a stability analysis for a linear scalar test case with periodic initial condition to find the best selection of APK polynomials and their specific modal filter. To the best of our knowledge, this work is the first that gives a stability analysis for a scheme with spectral filtering. Finally, we demonstrate the influence of the underlying basis of APK polynomials in a well-known test case.

References

Y. Allaneau and A. Jameson, Connections between the filtered discontinuous Galerkin method and the flux reconstruction approach to high order discretizations, Comput. Methods Appl. Mech. Engrg. 200 (2011), no. 49-52, 3628–3636. MR 2851584, DOI 10.1016/j.cma.2011.08.019
H. L. Atkins and C.-W. Shu, Quadrature-free implementation of discontinuous Galerkin method for hyperbolic equations, AIAA Journal 36 (1998), no. 5, 775–782.
G. E. Barter and D. L. Darmofal, Shock capturing with higher-order, PDE-based artificial viscosity, AIAA paper 3823 (2007).
M. G. Blyth and C. Pozrikidis, A Lobatto interpolation grid over the triangle, IMA J. Appl. Math. 71 (2006), no. 1, 153–169. MR 2203048, DOI 10.1093/imamat/hxh077
M. H. Carpenter and C. A. Kennedy, Fourth-order 2n-storage Runge-Kutta schemes, Nasa tm 109112 (1994), 871–885.
Moshe Dubiner, Spectral methods on triangles and other domains, J. Sci. Comput. 6 (1991), no. 4, 345–390. MR 1154903, DOI 10.1007/BF01060030
Charles F. Dunkl and Yuan Xu, Orthogonal polynomials of several variables, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 155, Cambridge University Press, Cambridge, 2014. MR 3289583, DOI 10.1017/CBO9781107786134
Anne Gelb and Jared Tanner, Robust reprojection methods for the resolution of the Gibbs phenomenon, Appl. Comput. Harmon. Anal. 20 (2006), no. 1, 3–25. MR 2200928, DOI 10.1016/j.acha.2004.12.007
J. Glaubitz, H. Ranocha, P. Öffner, and T. Sonar, Enhancing stability of correction procedure via reconstruction using summation-by-parts operators II: Modal filtering, 2016, Submitted.
D. Gottlieb and J. S. Hesthaven, Spectral methods for hyperbolic problems, J. Comput. Appl. Math. 128 (2001), no. 1-2, 83–131. Numerical analysis 2000, Vol. VII, Partial differential equations. MR 1820872, DOI 10.1016/S0377-0427(00)00510-0
David Gottlieb and Steven A. Orszag, Numerical analysis of spectral methods: theory and applications, CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1977. MR 520152
David Gottlieb and Chi-Wang Shu, On the Gibbs phenomenon and its resolution, SIAM Rev. 39 (1997), no. 4, 644–668. MR 1491051, DOI 10.1137/S0036144596301390
Jan S. Hesthaven and Robert M. Kirby, Filtering in Legendre spectral methods, Math. Comp. 77 (2008), no. 263, 1425–1452. MR 2398775, DOI 10.1090/S0025-5718-08-02110-8
P. G. Huang, Z. J. Wang, and Y. Liu, An implicit space-time spectral difference method for discontinuity capturing using adaptive polynomials, AIAA paper 5255 (2005).
H. T. Huynh, A flux reconstruction approach to high-order schemes including discontinuous Galerkin methods, AIAA paper 4079 (2007).
H. T. Huynh, High-order methods including discontinuous Galerkin by reconstructions on triangular meshes, AIAA Paper 44 (2011).
H. T. Huynh, Z. J. Wang, and P. E. Vincent, High-order methods for computational fluid dynamics: a brief review of compact differential formulations on unstructured grids, Comput. & Fluids 98 (2014), 209–220. MR 3209968, DOI 10.1016/j.compfluid.2013.12.007
Antony Jameson, A proof of the stability of the spectral difference method for all orders of accuracy, J. Sci. Comput. 45 (2010), no. 1-3, 348–358. MR 2679803, DOI 10.1007/s10915-009-9339-4
George Em Karniadakis and Spencer J. Sherwin, Spectral/$hp$ element methods for computational fluid dynamics, 2nd ed., Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2005. MR 2165335, DOI 10.1093/acprof:oso/9780198528692.001.0001
Robert M. Kirby and Spencer J. Sherwin, Aliasing errors due to quadratic nonlinearities on triangular spectral/$hp$ element discretisations, J. Engrg. Math. 56 (2006), no. 3, 273–288. MR 2292677, DOI 10.1007/s10665-006-9079-5
Tom Koornwinder, Two-variable analogues of the classical orthogonal polynomials, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Academic Press, New York-London, 1975, pp. 435–495. MR 402146
David A. Kopriva and John H. Kolias, A conservative staggered-grid Chebyshev multidomain method for compressible flows, J. Comput. Phys. 125 (1996), no. 1, 244–261. MR 1381812, DOI 10.1006/jcph.1996.0091
D. A. Kopriva and J. H. Kolias, A conservative staggered-grid Chebyshev multidomain method for compressible flow., Tech. report, DTIC Document, 1995.
Yen Liu, Marcel Vinokur, and Z. J. Wang, Spectral difference method for unstructured grids. I. Basic formulation, J. Comput. Phys. 216 (2006), no. 2, 780–801. MR 2235393, DOI 10.1016/j.jcp.2006.01.024
Heping Ma, Chebyshev-Legendre spectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal. 35 (1998), no. 3, 869–892. MR 1619902, DOI 10.1137/S0036142995293900
Yvon Maday, Sidi M. Ould Kaber, and Eitan Tadmor, Legendre pseudospectral viscosity method for nonlinear conservation laws, SIAM J. Numer. Anal. 30 (1993), no. 2, 321–342. MR 1211394, DOI 10.1137/0730016
G. May and A. Jameson, A spectral difference method for the Euler and Navier-Stokes equations on unstructured meshes, AIAA paper 304 (2006).
A. Meister, S. Ortleb, and Th. Sonar, Application of spectral filtering to discontinuous Galerkin methods on triangulations, Numer. Methods Partial Differential Equations 28 (2012), no. 6, 1840–1868. MR 2981873, DOI 10.1002/num.20705
Andreas Meister, Sigrun Ortleb, Thomas Sonar, and Martina Wirz, An extended discontinuous Galerkin and spectral difference method with modal filtering, ZAMM Z. Angew. Math. Mech. 93 (2013), no. 6-7, 459–464. MR 3069920, DOI 10.1002/zamm.201200051
Philipp Öffner and Thomas Sonar, Spectral convergence for orthogonal polynomials on triangles, Numer. Math. 124 (2013), no. 4, 701–721. MR 3073957, DOI 10.1007/s00211-013-0530-z
P. Öffner, T. Sonar, and M. Wirz, Detecting strength and location of jump discontinuities in numerical data, Applied Mathematics 4 (2013), no. 12, 1.
S. Ortleb, Ein diskontinuierliches Galerkin-Verfahren hoher Ordnung auf Dreiecksgittern mit modaler Filterung zur Lösung hyperbolischer Erhaltungsgleichungen, Kassel University Press GmbH, 2011.
P.-O. Persson and J. Peraire, Sub-cell shock capturing for discontinuous Galerkin methods, AIAA paper 112 (2006).
H. Ranocha, J. Glaubitz, P. Öffner, and T. Sonar, Enhancing stability of correction procedure via reconstruction using summation-by-parts operators I: Artificial dissipation, 2016, Submitted.
Hendrik Ranocha, Philipp Öffner, and Thomas Sonar, Summation-by-parts operators for correction procedure via reconstruction, J. Comput. Phys. 311 (2016), 299–328. MR 3461788, DOI 10.1016/j.jcp.2016.02.009
P. K. Suetin, Orthogonal polynomials in two variables, Analytical Methods and Special Functions, vol. 3, Gordon and Breach Science Publishers, Amsterdam, 1999. Translated from the 1988 Russian original by E. V. Pankratiev [E. V. Pankrat′ev]. MR 1717891
Yuzhi Sun, Z. J. Wang, and Yen Liu, High-order multidomain spectral difference method for the Navier-Stokes equations on unstructured hexahedral grids, Commun. Comput. Phys. 2 (2007), no. 2, 310–333. MR 2303929
Eitan Tadmor, Convergence of spectral methods for nonlinear conservation laws, SIAM J. Numer. Anal. 26 (1989), no. 1, 30–44. MR 977947, DOI 10.1137/0726003
Kris Van den Abeele, Chris Lacor, and Z. J. Wang, On the connection between the spectral volume and the spectral difference method, J. Comput. Phys. 227 (2007), no. 2, 877–885. MR 2442378, DOI 10.1016/j.jcp.2007.08.030
Kris Van den Abeele, Chris Lacor, and Z. J. Wang, On the stability and accuracy of the spectral difference method, J. Sci. Comput. 37 (2008), no. 2, 162–188. MR 2453217, DOI 10.1007/s10915-008-9201-0
Hervé Vandeven, Family of spectral filters for discontinuous problems, J. Sci. Comput. 6 (1991), no. 2, 159–192. MR 1140344, DOI 10.1007/BF01062118
P. E. Vincent, P. Castonguay, and A. Jameson, Insights from von Neumann analysis of high-order flux reconstruction schemes, J. Comput. Phys. 230 (2011), no. 22, 8134–8154. MR 2835413, DOI 10.1016/j.jcp.2011.07.013
P. E. Vincent, P. Castonguay, and A. Jameson, A new class of high-order energy stable flux reconstruction schemes, J. Sci. Comput. 47 (2011), no. 1, 50–72. MR 2781148, DOI 10.1007/s10915-010-9420-z
Z. J. Wang, High-order methods for the Euler and Navier–stokes equations on unstructured grids, Progress in Aerospace Sciences 43 (2007), no. 1, 1–41.
Z. J. Wang and Haiyang Gao, A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids, J. Comput. Phys. 228 (2009), no. 21, 8161–8186. MR 2573350, DOI 10.1016/j.jcp.2009.07.036
Z. J. Wang, Yen Liu, Georg May, and Antony Jameson, Spectral difference method for unstructured grids. II. Extension to the Euler equations, J. Sci. Comput. 32 (2007), no. 1, 45–71. MR 2320559, DOI 10.1007/s10915-006-9113-9
M. Wirz, Ein Spektrale-Differenzen-Verfahren mit modaler Filterung und zweidimensionaler Kantendetektierung mithilfe konjugierter Fourierreihen, Cuvillier, 2012.
M. Wirz, Detecting edges in high order methods for hyperbolic conservation laws, High Order Nonlinear Numerical Schemes for Evolutionary PDEs, Springer, 2014, pp. 151–167.
Meilin Yu, Z. J. Wang, and Yen Liu, On the accuracy and efficiency of discontinuous Galerkin, spectral difference and correction procedure via reconstruction methods, J. Comput. Phys. 259 (2014), 70–95. MR 3148560, DOI 10.1016/j.jcp.2013.11.023