Abstract
We give a brief review of our recently proposed WENO finite volume method for Cartesian grids (Buchmüller, Helzel, J Sci Comput, 61:343–368, 2014, [3]), (Buchmüller et al. Appl Math Comput, 272:460–478, 2016, [4]), which increases the accuracy of the commonly used so-called dimension-by-dimension approach. The main ingredient is a transformation from high-order accurate interface-averaged values of the conserved quantities to high-order accurate point values. Using these point values of the conserved quantities, we can compute point values of the fluxes at the center of each grid cell interface. Finally, the reverse transformation is used in order to compute high-order accurate face-averaged numerical fluxes that can be used in a finite volume method. While we have recently concentrated on spatially two-dimensional test calculations, we will here present a new three-dimensional test problem, which confirms the accuracy and efficiency of our approach. This test problem, a rotated vortex, should be of general interest to researchers who want to test the accuracy of their three-dimensional schemes.
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
R. Borges, M. Carmona, B. Costa, W.S. Don, An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3191–3211 (2008)
P. Buchmüller, High-Order WENO Finite Volume Methods on Cartesian Grids with Adaptive Mesh Refinement, Ph.D. thesis, Heinrich-Heine-University Düsseldorf (2016)
P. Buchmüller, C. Helzel, Improved accuracy of high-order WENO finite volume methods on Cartesian grids. J. Sci. Comput. 61, 343–368 (2014)
P. Buchmüller, J. Dreher, C. Helzel, Finite volume WENO methods for hyperbolic conservation laws on Cartesian grids with adaptive mesh refinement. Appl. Math. Comput. 272, 460–478 (2016)
J. Casper, H. Atkins, A finite-volume high-order ENO scheme for two-dimensional hyperbolic systems. J. Comput. Phys. 106, 62–76 (1993)
P. Colella, M. Dorr, J. Hittinger, D. Martin, High-order finite volume methods in mapped coordinates. J. Comput. Phys. 230, 2952–2976 (2011)
W.-S. Don, R. Borges, Accuracy of the weighted essentially non-oscillatory conservative finite difference schemes. J. Comput. Phys. 250, 347–372 (2013)
J. Dreher, R. Grauer, Racoon: a parallel mesh-adaptive framework for hyperbolic conservation laws. Parallel Comput. 31, 913–932 (2005)
S.Y. Kadioglu, R. Klein, M.L. Minion, A fourth-order auxiliary variable projection method for zero-Mach number gas dynamics. J. Comput. Phys. 227, 2012–2043 (2008)
M. Parsani, D.I. Ketcheson, W. Deconick, Optimized explicit Runge-Kutta schemes for the spectral difference method applied to wave propagation problems. SIAM J. Sci. Comput. 35, A957–A986 (2013)
C.-W. Shu, High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51, 82–126 (2009)
R. Zhang, M. Zhang, C.-W. Shu, On the order of accuracy and numerical performance of two classes of finite volume WENO schemes. Commun. Comput. Phys. 9, 807–827 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Buchmüller, P., Dreher, J., Helzel, C. (2018). Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids with Adaptive Mesh Refinement. In: Klingenberg, C., Westdickenberg, M. (eds) Theory, Numerics and Applications of Hyperbolic Problems I. HYP 2016. Springer Proceedings in Mathematics & Statistics, vol 236. Springer, Cham. https://doi.org/10.1007/978-3-319-91545-6_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-91545-6_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-91544-9
Online ISBN: 978-3-319-91545-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)