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Abstract In this article, we study the spectral volume (SV) methods for scalar hyperbolic conservation laws with a class of subdivision points under the Petrov–Galerkin framework. Due to the strong connection between the DG method and the SV method with the appropriate choice of the subdivision points, it is natural to analyze the SV method in the Galerkin form and derive the analogous theoretical results as in the DG method. This article considers a class of SV methods, whose subdivision points are the zeros of a specific polynomial with a parameter in it. Properties of the piecewise constant functions under this subdivision, including the orthogonality between the trial solution space and test function space, are provided. With the aid of these properties, we are able to derive the energy stability, optimal a priori error estimates of SV methods with arbitrary high order accuracy. We also study the superconvergence of the numerical solution with the correction function technique, and show the order of superconvergence would be different with different choices of the subdivision points. In the numerical experiments, by choosing different parameters in the SV method, the theoretical findings are confirmed by the numerical results.
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Cremona Orbits in P^4 and Applications
This article is motivated by the authors’ interest in the geometry of the Mori dream space P^4 blown up in 8 general points. In this article, we develop the necessary technique for determining Weyl orbits of linear cycles for the four-dimensional case, by explicit computations in the Chow ring of the resolution of the standard Cremona transformation. In particular, we close this paper with applications to the question of the dimension of the space of global sections of effective divisors having at most 8 points.
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- Award ID(s):
- 2152130
- PAR ID:
- 10514985
- Publisher / Repository:
- Birkhauser, Springer
- Date Published:
- Journal Name:
- Trends in Mathematics
- ISSN:
- 2297-0215
- ISBN:
- 978-3-031-11938-5
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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