隐式迎风有限体积方案的分析：具有粗糙系数的情况
Analysis of the implicit upwind finite volume scheme with rough coefficients
AI总结：
本文研究了在低正则性DiPerna-Lions设定下，隐式迎风有限体积方案对线性连续方程数值近似的收敛性，证明了在无结构规则网格上，近似解向连续模型唯一分布解收敛的速率至少为1/2，并通过对数Kantorovich-Rubinstein距离估计了数值误差，从而给出了弱收敛速率的界。
摘要：
我们研究了隐式迎风有限体积方案对线性连续方程在低正则性DiPerna-Lions设定下的数值近似。也就是说，我们关注的是空间Sobolev正则的速度场以及仅可积的数据。我们证明，在无结构规则网格上，由迎风方案生成的近似解向连续模型唯一分布解收敛的速率至少为1/2。数值误差以对数Kantorovich-Rubinstein距离来估计，从而给出了弱收敛速率的界。
英文摘要：
We study the implicit upwind finite volume scheme for numerically approximating the linear continuity equation 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 prove that on unstructured regular meshes the rate of convergence of approximate solutions generated by the upwind scheme towards the unique distributional solution of the continuous model is at least 1/2. The numerical error is estimated in terms of logarithmic Kantorovich-Rubinstein distances and provides thus a bound on the rate of weak convergence.
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