Structure preservation in high-order hybrid discretisations of potential-driven advection-diffusion: linear and nonlinear approaches

Simon Lemaire; Julien Moatti; Simon Lemaire; Julien Moatti

doi:10.3934/mine.2024005

Mathematics in Engineering

2024, Volume 6, Issue 1: 100-136. doi: 10.3934/mine.2024005

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Structure preservation in high-order hybrid discretisations of potential-driven advection-diffusion: linear and nonlinear approaches

Simon Lemaire ^{1
,
,},
Julien Moatti ^1,2

1.
Inria, Univ. Lille, CNRS, UMR 8524–Laboratoire Paul Painlevé, F-59000 Lille, France
2.
Institute of Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8–10, A-1040 Wien, Austria

Received: 19 October 2023 Revised: 12 January 2024 Accepted: 17 January 2024 Published: 24 January 2024

We are interested in the high-order approximation of anisotropic, potential-driven advection-diffusion models on general polytopal partitions. We study two hybrid schemes, both built upon the Hybrid High-Order technology. The first one hinges on exponential fitting and is linear, whereas the second is nonlinear. The existence of solutions is established for both schemes. Both schemes are also shown to possess a discrete entropy structure, ensuring that the long-time behaviour of discrete solutions mimics the PDE one. For the nonlinear scheme, the positivity of discrete solutions is a built-in feature. On the contrary, we display numerical evidence indicating that the linear scheme violates positivity, whatever the order. Finally, we verify numerically that the nonlinear scheme has optimal order of convergence, expected long-time behaviour, and that raising the polynomial degree results, also in the nonlinear case, in an efficiency gain.
- high-order methods,
- hybrid methods,
- polytopal meshes,
- structure-preserving schemes,
- potential-driven advection-diffusion,
- long-time behaviour,
- entropy methods
Citation: Simon Lemaire, Julien Moatti. Structure preservation in high-order hybrid discretisations of potential-driven advection-diffusion: linear and nonlinear approaches[J]. Mathematics in Engineering, 2024, 6(1): 100-136. doi: 10.3934/mine.2024005

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Abstract

We are interested in the high-order approximation of anisotropic, potential-driven advection-diffusion models on general polytopal partitions. We study two hybrid schemes, both built upon the Hybrid High-Order technology. The first one hinges on exponential fitting and is linear, whereas the second is nonlinear. The existence of solutions is established for both schemes. Both schemes are also shown to possess a discrete entropy structure, ensuring that the long-time behaviour of discrete solutions mimics the PDE one. For the nonlinear scheme, the positivity of discrete solutions is a built-in feature. On the contrary, we display numerical evidence indicating that the linear scheme violates positivity, whatever the order. Finally, we verify numerically that the nonlinear scheme has optimal order of convergence, expected long-time behaviour, and that raising the polynomial degree results, also in the nonlinear case, in an efficiency gain.

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