Abstract
The Peaceman–Rachford alternating direction implicit (ADI) method is considered for the time-integration of a class of wave-type equations for linear, isotropic materials on a tensorial domain, e.g., a cuboid in 3D or a rectangle in 2D. This method is known to be unconditionally stable and of conventional order two. So far, it has been applied to specific problems and is mostly combined with finite differences in space, where it can be implemented at the cost of an explicit method.
In this paper, we consider the ADI method for a discontinuous Galerkin (dG) space discretization. We characterize a large class of first-order differential equations for which we show that on tensorial meshes, the method can be implemented with optimal (linear) complexity.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
W. Bangerth, R. Hartmann, G. Kanschat, deal.II – a General Purpose Object Oriented Finite Element Library, ACM Trans. Math. Softw. 33(4), 24/1–24/27 (2007)
D.A. Di Pietro, A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Springer Mathematics and Applications (Springer, Berlin, 2012)
J. S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods. Springer Texts in Applied Mathematics (Springer, Berlin, 2008)
M. Hochbruck, T. Jahnke, R. Schnaubelt, Convergence of an ADI splitting for Maxwell’s equations. Numer. Math. 129, 535–561 (2015)
T. Namiki, A new FDTD algorithm based on alternating-direction implicit method. IEEE Trans. Microwave Theory Tech. 47, 2003–2007 (1999)
D.W. Peaceman, H.H. Rachford Jr., The numerical solution of parabolic and elliptic differential equations. J. Soc. Ind. Appl. Math. 3, 28–41 (1955)
F. Zhen, Z. Chen, J. Zhang, Toward the development of a three-dimensional unconditionally stable finite-difference time-domain method. IEEE Trans. Microwave Theory Tech. 48, 1550–1558 (2000)
Acknowledgements
We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Hochbruck, M., Köhler, J. (2019). On the Efficiency of the Peaceman–Rachford ADI-dG Method for Wave-Type Problems. In: Radu, F., Kumar, K., Berre, I., Nordbotten, J., Pop, I. (eds) Numerical Mathematics and Advanced Applications ENUMATH 2017. ENUMATH 2017. Lecture Notes in Computational Science and Engineering, vol 126. Springer, Cham. https://doi.org/10.1007/978-3-319-96415-7_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-96415-7_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-96414-0
Online ISBN: 978-3-319-96415-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)