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

TL;DR

This work develops and analyzes two high-order Hybrid High-Order schemes for anisotropic, potential-driven advection-diffusion on polytopal meshes. A linear exponential-fitting scheme and a nonlinear logarithmic scheme are shown to exist, preserve a discrete entropy structure, and conserve mass, with the nonlinear scheme additionally guaranteeing positivity and matching long-time equilibrium. Numerical experiments demonstrate exponential convergence to equilibrium, highlight positivity violations in linear schemes, and reveal substantial efficiency gains from high-order nonlinear discretizations. The results support the use of nonlinear high-order HHO methods for dissipative drift-diffusion-type problems and motivate further theoretical and practical extensions to nonlinear and semiconductor models.

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.

Figures (5)

• Figure 1: Accuracy vs. mesh size. Relative errors on triangular meshes.
• Figure 2: Accuracy vs. computational cost. Relative errors on triangular meshes.
• Figure 3: Accuracy on general meshes. Relative $L^2_t(L^2_x)$-error on distorted quadrangular meshes.
• Figure 4: Long-time behaviour of discrete solutions.$L^1$-distance to $u^\infty$ on Kershaw meshes.
• Figure 5: Long-time behaviour of discrete solutions.$L^1$-distance to $u^\infty$ on distorted quadrangular meshes.

Theorems & Definitions (28)

• Definition 1: Preservation of the thermal equilibrium
• Remark 1: Relaxation of the mesh regularity assumptions
• Remark 2: Non-polynomial definition
• Remark 3: Initial condition
• Remark 4: Alternative scheme definition
• Remark 5: Parameter $\varepsilon$
• Remark 6: Initial condition
• Remark 7: Lowest-order versions of the schemes ($k=0$)
• Proposition 1: Preservation of the thermal equilibrium
• Proposition 2: Well-posedness of the exponential fitting scheme