{"4b6636216b66dfdfd50ca53bb93685demathematik":{"DOI":"10.1002/num.22180","ISBN":"","ISSN":"","URL":"https://onlinelibrary.wiley.com/doi/abs/10.1002/num.22180","abstract":"This article concerns with incorporating wavelet bases into existing\n\tstreamline upwind Petrov-Galerkin (SUPG) methods for the numerical\n\tsolution of nonlinear hyperbolic conservation laws which are known\n\tto develop shock solutions. Here, we utilize an SUPG formulation\n\tusing continuous Galerkin in space and discontinuous Galerkin in\n\ttime. The main motivation for such a combination is that these methods\n\thave good stability properties thanks to adding diffusion in the\n\tdirection of streamlines. But they are more expensive than explicit\n\tsemidiscrete methods as they have to use space-time formulations.\n\tUsing wavelet bases we maintain the stability properties of SUPG\n\tmethods while we reduce the cost of these methods significantly through\n\tnatural adaptivity of wavelet expansions. In addition, wavelet bases\n\thave a hierarchical structure. We use this property to numerically\n\tinvestigate the hierarchical addition of an artificial diffusion\n\tfor further stabilization in spirit of spectral diffusion. Furthermore,\n\twe add the hierarchical diffusion only in the vicinity of discontinuities\n\tusing the feature of wavelet bases in detection of location of discontinuities.\n\tAlso, we again use the last feature of the wavelet bases to perform\n\ta postprocessing using a denosing technique based on a minimization\n\tformulation to reduce Gibbs oscillations near discontinuities while\n\tkeeping other regions intact. Finally, we show the performance of\n\tthe proposed combination through some numerical examples including\n\tBurgers�, transport, and wave equations as well as systems of shallow\n\twater equations.� 2017 Wiley Periodicals, Inc. Numer Methods Partial\n\tDifferential Eq 33: 2062�2089, 2017","annote":"","author":[{"family":"Minbashian","given":"Hadi"},{"family":"Adibi","given":"Hojatolah"},{"family":"Dehghan","given":"Mehdi"}],"citation-label":"minbashian2017adaptive","collection-editor":[],"collection-title":"","container-author":[],"container-title":"Numerical Methods for Partial Differential Equations","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2017"]],"literal":"2017"},"event-place":"","id":"4b6636216b66dfdfd50ca53bb93685demathematik","interhash":"c8972e3ab25e760174ed7741330758be","intrahash":"4b6636216b66dfdfd50ca53bb93685de","issue":"6","issued":{"date-parts":[["2017"]],"literal":"2017"},"keyword":"(SUPG), Galerkin, Petrove-Galerkin adaptive conservation continuous discontinuous from:mhartmann hyperbolic ians laws, method, postprocessing spectral streamline upwind viscosity, vorlaeufig wavelet","misc":{"owner":"seusdd","doi":"10.1002/num.22180","eprint":"https://onlinelibrary.wiley.com/doi/pdf/10.1002/num.22180"},"note":"","number":"6","number-of-pages":"27","page":"2062-2089","page-first":"2062","publisher":"","publisher-place":"","status":"","title":"An adaptive wavelet space-time SUPG method for hyperbolic conservation\n\tlaws","type":"article-journal","username":"mathematik","version":"","volume":"33"},"4b6636216b66dfdfd50ca53bb93685demhartmann":{"DOI":"10.1002/num.22180","ISBN":"","ISSN":"","URL":"https://onlinelibrary.wiley.com/doi/abs/10.1002/num.22180","abstract":"This article concerns with incorporating wavelet bases into existing\n\tstreamline upwind Petrov-Galerkin (SUPG) methods for the numerical\n\tsolution of nonlinear hyperbolic conservation laws which are known\n\tto develop shock solutions. Here, we utilize an SUPG formulation\n\tusing continuous Galerkin in space and discontinuous Galerkin in\n\ttime. The main motivation for such a combination is that these methods\n\thave good stability properties thanks to adding diffusion in the\n\tdirection of streamlines. But they are more expensive than explicit\n\tsemidiscrete methods as they have to use space-time formulations.\n\tUsing wavelet bases we maintain the stability properties of SUPG\n\tmethods while we reduce the cost of these methods significantly through\n\tnatural adaptivity of wavelet expansions. In addition, wavelet bases\n\thave a hierarchical structure. We use this property to numerically\n\tinvestigate the hierarchical addition of an artificial diffusion\n\tfor further stabilization in spirit of spectral diffusion. Furthermore,\n\twe add the hierarchical diffusion only in the vicinity of discontinuities\n\tusing the feature of wavelet bases in detection of location of discontinuities.\n\tAlso, we again use the last feature of the wavelet bases to perform\n\ta postprocessing using a denosing technique based on a minimization\n\tformulation to reduce Gibbs oscillations near discontinuities while\n\tkeeping other regions intact. Finally, we show the performance of\n\tthe proposed combination through some numerical examples including\n\tBurgers�, transport, and wave equations as well as systems of shallow\n\twater equations.� 2017 Wiley Periodicals, Inc. Numer Methods Partial\n\tDifferential Eq 33: 2062�2089, 2017","annote":"","author":[{"family":"Minbashian","given":"Hadi"},{"family":"Adibi","given":"Hojatolah"},{"family":"Dehghan","given":"Mehdi"}],"citation-label":"minbashian2017adaptive","collection-editor":[],"collection-title":"","container-author":[],"container-title":"Numerical Methods for Partial Differential Equations","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2017"]],"literal":"2017"},"event-place":"","id":"4b6636216b66dfdfd50ca53bb93685demhartmann","interhash":"c8972e3ab25e760174ed7741330758be","intrahash":"4b6636216b66dfdfd50ca53bb93685de","issue":"6","issued":{"date-parts":[["2017"]],"literal":"2017"},"keyword":"(SUPG), Galerkin, Petrove-Galerkin adaptive conservation continuous discontinuous hyperbolic laws, method, postprocessing spectral streamline upwind viscosity, vorlaeufig wavelet","misc":{"owner":"seusdd","doi":"10.1002/num.22180","eprint":"https://onlinelibrary.wiley.com/doi/pdf/10.1002/num.22180"},"note":"","number":"6","number-of-pages":"27","page":"2062-2089","page-first":"2062","publisher":"","publisher-place":"","status":"","title":"An adaptive wavelet space-time SUPG method for hyperbolic conservation\n\tlaws","type":"article-journal","username":"mhartmann","version":"","volume":"33"}}