We study the implicit upwind finite volume scheme for numerically approximating the advection-diffusion equation with a vector field in the low regularity DiPerna-Lions setting. That is, we are concerned with advecting velocity fields that are spatially Sobolev regular and data that are merely integrable. We study the implicit upwind finite volume scheme for numerically approximating the advection-diffusion equation with a vector field in the low regularity DiPerna-Lions setting. We prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique solution of the continuous model is at least one. The numerical error is estimated in terms of logarithmic Kantorovich-Rubinstein distances and provides thus a bound on the rate of weak convergence.

翻译：我们研究在低正则性DiPerna-Lions框架下，针对具有向量场的对流扩散方程数值逼近的隐式迎风有限体积格式。即，我们关注空间Sobolev正则的平流速度场和仅可积的数据。我们证明了在非结构正则网格上，由迎风格式生成的近似解向连续模型唯一解的收敛速率至少为一。数值误差通过对数Kantorovich-Rubinstein距离进行估计，由此给出了弱收敛速率的界。