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.

翻译：本文关注各向异性势驱动对流-扩散模型在一般多面体剖分上的高阶近似问题。我们研究了两种基于混合高阶技术的格式：第一种采用指数拟合的线性格式，第二种为非线性格式。两种格式均建立了解的存在性理论，并证明其具备离散熵结构，从而确保离散解的长时间行为与偏微分方程解的行为一致。对于非线性格式，离散解的正性是其固有特征；反之，数值实验表明线性格式无论阶次如何均会破坏正性。最后通过数值验证：非线性格式具有最优收敛阶、预期的长时间行为，且提升多项式次数（包括非线性情形）可带来效率增益。