We are interested in the high-order approximation of anisotropic advection-diffusion problems 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 enjoy a discrete entropy structure, ensuring that the discrete long-time behaviour of solutions mimics the PDE one. The nonlinear scheme is designed so as to enforce the positivity of discrete solutions. 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：本文关注一般多边形剖分上各向异性对流-扩散问题的高阶逼近。我们研究了两种均基于高阶混合技术的混合格式：第一种采用指数拟合并保持线性特性，第二种则为非线性格式。两种格式均建立了解的存在性理论，并证明它们具有离散熵结构，确保解的长期离散行为与偏微分方程模型一致。非线性格式通过设计强制保证离散解的正定性。与之相反，数值实验表明，无论阶次如何，线性格式均会破坏正定性。最后，数值验证显示非线性格式具有最优收敛阶、预期的长期行为，且提高多项式次数在非线性情形下同样能提升计算效率。