Advertisement

Stability of Higher-Order ALE-STDGM for Nonlinear Problems in Time-Dependent Domains

  • Monika BalázsováEmail author
  • Miloslav Vlasák
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)

Abstract

In this paper we investigate the stability of the space-time discontinuous Galerkin method for the solution of nonstationary, nonlinear convection-diffusion problem in time-dependent domains. At first we define the continuous problem and reformulate it using the Arbitrary Lagrangian-Eulerian (ALE) method, which replaces the classical partial time derivative by the so called ALE-derivative and an additional convective term. Then the problem is discretized with the aid of the ALE space-time discontinuous Galerkin method (ALE-STDGM). The discretization uses piecewise polynomial functions of degree p ≥ 1 in space and q > 1 in time. Finally in the last part of the paper we present our results concerning the unconditional stability of the method. An important step is the generalization of a discrete characteristic function associated with the approximate solution and the derivation of its properties, namely its continuity in the \(\Vert \cdot \Vert _{L^2}\)-norm and in special ∥⋅∥DG-norm.

Notes

Acknowledgements

This research was supported by the project GA UK No. 127615 of the Charles University (M. Balázsová) and by the grant 17-01747S of the Czech Science Foundation (M. Vlasák, who is a junior member of the University Centre for Mathematical Modeling, Applied Analysis and Computational Mathematics - MathMAC).

References

  1. 1.
    M. Balázsová, M. Feistauer, On the stability of the space-time discontinuous Galerkin method for nonlinear convection-diffusion problems in time-dependent domains. Appl. Math. 60, 501–526 (2015)MathSciNetCrossRefGoogle Scholar
  2. 2.
    M. Balázsová, M. Feistauer, On the uniform stability of the space-time discontinuous Galerkin method for nonstationary problems in time-dependent domains, in ALGORITMY 2016, 20th Conference on Scientific Computing, Vysoké Tatry - Podbanské, Slovakia, March 13–18, 2016, ed. by A. Handlovičová, D. Ševčovič (Slovak University of Technology, Bratislava, 2016), pp. 84–92Google Scholar
  3. 3.
    J. Česenek, M. Feistauer, J. Horáček, V. Kučera, J. Prokopová, Simulation of compressible viscous flow in time-dependent domains. Appl. Math. Comput. 219, 7139–7150 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    J. Česenek, M. Feistauer, A. Kosík, DGFEM for the analysis of airfoil vibrations induced by compressible flow. ZAMM Z. Angew. Math. Mech. 93(6–7), 387–402 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    V. Dolejší, M. Feistauer, Discontinuous Galerkin method – Analysis and Applications to Compressible Flow (Springer, Berlin, 2015)zbMATHGoogle Scholar
  6. 6.
    A. Kosík, M. Feistauer, M. Hadrava, J. Horáček, Numerical simulation of the interaction between a nonlinear elastic structure and compressible flow by the discontinuous Galerkin method. Appl. Math. Comput. 267, 382–396 (2015)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Faculty of Mathematics and PhysicsCharles UniversityPraha 8Czech Republic

Personalised recommendations