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Mathematics of Computation

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Application of modal filtering to a spectral difference method


Authors: Jan Glaubitz, Philipp Öffner and Thomas Sonar
Journal: Math. Comp. 87 (2018), 175-207
MSC (2010): Primary 65M12, 65M70; Secondary 42C10
DOI: https://doi.org/10.1090/mcom/3257
Published electronically: August 7, 2017
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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.


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Additional Information

Jan Glaubitz
Affiliation: Technische Universität Braunschweig, Institut Computational Mathematics, Pockelsstraße 14, D-38106 Braunschweig, Germany
Email: j.glaubitz@tu-bs.de

Philipp Öffner
Affiliation: Technische Universität Braunschweig, Institut Computational Mathematics, Pockelsstraße 14, D-38106 Braunschweig, Germany
Email: p.oeffner@tu-bs.de

Thomas Sonar
Affiliation: Technische Universität Braunschweig, Institut Computational Mathematics, Pockelsstraße 14, D-38106 Braunschweig, Germany
Email: t.sonar@tu-bs.de

DOI: https://doi.org/10.1090/mcom/3257
Keywords: Hyperbolic conservation laws, high order methods, Spectral Difference Method, orthogonal polynomials, modal filtering
Received by editor(s): April 2, 2016
Received by editor(s) in revised form: September 13, 2016
Published electronically: August 7, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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