Abstract
Filtering is necessary to stabilize piecewise smooth solutions. The resulting diffusion stabilizes the method, but may fail to resolve the solution near discontinuities. Moreover, high order filtering still requires cost prohibitive time stepping. This paper introduces an adaptive filter that controls spurious modes of the solution, but is not unnecessarily diffusive. Consequently we are able to stabilize the solution with larger time steps, but also take advantage of the accuracy of a high order filter.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
A. Gelb, E. Tadmor, Enhanced spectral viscosity approximations for conservation laws. Appl. Numer. Math. 33, 3–21 (2000)
J.S. Hesthaven, S. Gottlieb, D. Gottlieb, Spectral Methods for Time-Dependent Problems (Cambridge University Press, Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo, 2007)
R.J. Leveque, Finite Volume Methods for Hyperbolic Problems (Cambridge University Press, Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo, Delhi, Tokyo, Mexico City, 2002)
Y. Li, S. Osher, Coordinate descent optimization for l 1 minimization with application to compressed sensing; a Greedy algorithm. Inverse Prob. Imaging 3(3), 487–503 (2009)
S.C. Reddy, L.N. Trefethen, Stability of the method of lines. Numer. Math. 62, 235–267 (1992)
H. Schaeffer, R. Caflisch, C. Hauck, S. Osher, Sparse dynamics for partial differential equations. Proc. Natl. Acad. Sci. USA 110(17), 6634–6639 (2013)
J.C. Schatzman, Accuracy of the discrete fourier transform and the fast fourier transform. SIAM J. Sci. Comput. 17(5), 1150–1166 (1996)
Acknowledgements
The works of Dennis Denker and Anne Gelb are supported in part by grants NSF-DMS 1216559 and AFOSR FA9550-12-1-0393.
The submitted manuscript is based upon work of Rick Archibald, authored in part by contractors [UT-Battelle LLC, manager of Oak Ridge National Laboratory (ORNL)], and supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Contract No. DE-AC05-00OR22725.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Denker, D., Archibald, R., Gelb, A. (2015). An Adaptive Fourier Filter for Relaxing Time Stepping Constraints for Explicit Solvers. In: Kirby, R., Berzins, M., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations ICOSAHOM 2014. Lecture Notes in Computational Science and Engineering, vol 106. Springer, Cham. https://doi.org/10.1007/978-3-319-19800-2_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-19800-2_12
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-19799-9
Online ISBN: 978-3-319-19800-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)