Abstract
In this paper, we stabilize and limit continuous Galerkin discretizations of a linear transport equation using an algebraic approach to derivation of artificial diffusion operators. Building on recent advances in the analysis and design of edge-based algebraic flux correction schemes for singularly perturbed convection-diffusion problems, we derive algebraic stabilization operators that generate nonlinear high-order stabilization in smooth regions and enforce discrete maximum principles everywhere. The correction factors for antidiffusive element or edge contributions are defined in terms of nodal gradients that vanish at local extrema. The proposed limiting strategy is linearity-preserving and provides Lipschitz continuity of constrained terms. Numerical examples are presented for two-dimensional test problems.
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
Abgrall, R.: High order schemes for hyperbolic problems using globally continuous approximation and avoiding mass matrices. J. Sci. Comput. 73, 461–494 (2017)
Badia, S., Bonilla, J.: Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization. Comput. Methods Appl. Mech. Eng. 313, 133–158 (2017)
Barrenechea, G., John, V., Knobloch, P.: Analysis of algebraic flux correction schemes. SIAM J. Numer. Anal. 54, 2427–2451 (2016)
Barrenechea, G., John, V., Knobloch, P.: A linearity preserving algebraic flux correction scheme satisfying the discrete maximum principle on general meshes. Math. Models Methods Appl. Sci. 27, 525–548 (2017)
Barrenechea, G., John, V., Knobloch, P., Rankin, R.: A unified analysis of algebraic flux correction schemes for convection-diffusion equations. SeMA (2018). https://doi.org/10.1007/s40324-018-0160-6
Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)
Guermond, J.-L., Nazarov, M., Popov, B., Yang, Y.: A second-order maximum principle preserving Lagrange finite element technique for nonlinear scalar conservation equations. SIAM J. Numer. Anal. 52, 2163–2182 (2014)
Jameson, A.: Positive schemes and shock modelling for compressible flows. Int. J. Numer. Methods Fluids 20, 743–776 (1995)
Kuzmin, D.: Algebraic flux correction I. Scalar conservation laws. In: Kuzmin, D., Löhner, R., Turek, S. (eds.) Flux-Corrected Transport: Principles, Algorithms, and Applications, 2nd edn., pp. 145–192. Springer, Dordrecht (2012)
Kuzmin, D., Basting, S., Shadid, J.N.: Linearity-preserving monotone local projection stabilization schemes for continuous finite elements. Comput. Methods Appl. Mech. Eng. 322, 23–41 (2017)
Lohmann, C.: Algebraic flux correction schemes preserving the eigenvalue range of symmetric tensor fields. Ergebnisber. Inst. Angew. Math. 584, TU Dortmund University (2018)
Lohmann, C., Kuzmin, D., Shadid, J.N., Mabuza, S.: Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements. J. Comput. Phys. 344, 151–186 (2017)
Neta, B., Williams, R.T.: Stability and phase speed for various finite element formulations of the advection equation. Comput. Fluids 14, 393–410 (1986)
Acknowledgements
This research was supported by the German Research Association (DFG) under grant KU 1530/23-1. The author would like to thank Christoph Lohmann (TU Dortmund University) for helpful discussions and suggestions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Kuzmin, D. (2020). Gradient-Based Limiting and Stabilization of Continuous Galerkin Methods. In: van Brummelen, H., Corsini, A., Perotto, S., Rozza, G. (eds) Numerical Methods for Flows. Lecture Notes in Computational Science and Engineering, vol 132. Springer, Cham. https://doi.org/10.1007/978-3-030-30705-9_29
Download citation
DOI: https://doi.org/10.1007/978-3-030-30705-9_29
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-30704-2
Online ISBN: 978-3-030-30705-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)