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:我们用低常规 DiPerna-Lions 设置的矢量字段来研究在数字上近似平面反向扩散方程式的隐性上风有限体积方案。也就是说,我们关注的是空间上的Sobolev 常规速度字段和数据仅是无法识别的数据。我们研究的是在低常规 DiPerna-Lions 设置的矢量字段中,在数字上风反向扩散方程式与矢量字段相近的隐性有限体积方案。我们证明,在非结构正常的中间,上风方案产生的接近连续模型独特解决方案的趋同率至少为1。数字错误用对数 Kantorovich-Rubinstein 距离来估计,从而根据衰弱的趋同率进行约束。