On the full and global accuracy of a compact third order WENO scheme. II. (English) Zbl 1367.65119
Karasözen, Bülent (ed.) et al., Numerical mathematics and advanced applications – ENUMATH 2015. Selected papers based on the presentations at the European conference, Ankara, Turkey, September 14–18, 2015. Cham: Springer (ISBN 978-3-319-39927-0/hbk; 978-3-319-39929-4/ebook). Lecture Notes in Computational Science and Engineering 112, 53-62 (2016).
Summary: Recently in [SIAM J. Numer. Anal. 52, No. 5, 2335–2355 (2014; Zbl 1408.65062)], the author showed for which parameter range the compact third order WENO reconstruction procedure introduced in [D. Levy et al., SIAM J. Sci. Comput. 22, No. 2, 656–672 (2000; Zbl 0967.65089)] reaches the optimal order of accuracy (\(h^{3}\) in the smooth case and \(h^{2}\) near discontinuities). This is the case for the parameter choice \(\varepsilon = Kh^q\) in the weight design with \(q\leq 3\) and \(pq\geq 2\), where \(p\geq 1\) is the exponent used in the computation of the weights in the WENO scheme. While these theoretical results for the convergence rates of the WENO reconstruction procedure could also be validated in the numerical tests, the application within the semi-discrete central scheme of A. Kurganov and D. Levy [ibid. 22, No. 4, 1461–1488 (2000; Zbl 0979.65077)] together with a third order TVD-Runge-Kutta scheme for the time integration did not yield a third order accurate scheme in total for \(q > 2\). The aim of this follow-up paper is to explain this observation with further analytical and numerical results.
For the entire collection see [Zbl 1358.65003].
For the entire collection see [Zbl 1358.65003].
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35L65 | Hyperbolic conservation laws |
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