Advertisement

On the Efficiency of the Peaceman–Rachford ADI-dG Method for Wave-Type Problems

  • Marlis Hochbruck
  • Jonas KöhlerEmail author
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

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.

Notes

Acknowledgements

We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) through CRC 1173.

References

  1. 1.
    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)Google Scholar
  2. 2.
    D.A. Di Pietro, A. Ern, Mathematical Aspects of Discontinuous Galerkin Methods. Springer Mathematics and Applications (Springer, Berlin, 2012)CrossRefGoogle Scholar
  3. 3.
    J. S. Hesthaven, T. Warburton, Nodal Discontinuous Galerkin Methods. Springer Texts in Applied Mathematics (Springer, Berlin, 2008)Google Scholar
  4. 4.
    M. Hochbruck, T. Jahnke, R. Schnaubelt, Convergence of an ADI splitting for Maxwell’s equations. Numer. Math. 129, 535–561 (2015)MathSciNetCrossRefGoogle Scholar
  5. 5.
    T. Namiki, A new FDTD algorithm based on alternating-direction implicit method. IEEE Trans. Microwave Theory Tech. 47, 2003–2007 (1999)CrossRefGoogle Scholar
  6. 6.
    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)MathSciNetCrossRefGoogle Scholar
  7. 7.
    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)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Karlsruhe Institute of TechnologyKarlsruheGermany

Personalised recommendations