带粗糙系数的对流扩散方程有限体积格式的误差估计
Error estimates for a finite volume scheme for advection-diffusion equations with rough coefficients
AI总结：
该研究针对低正则性DiPerna-Lions框架下带粗糙系数的对流扩散方程，分析隐式迎风有限体积格式的收敛性，证明非结构正则网格上收敛速率至少为一阶，并通过对数Kantorovich-Rubinstein距离给出弱收敛误差界。
摘要：
我们研究低正则性DiPerna-Lions（迪佩尔纳-利翁斯）框架下、带向量场的对流扩散方程数值逼近的隐式迎风格式有限体积法。即我们关注的是空间上具Sobolev（索伯列夫）正则性的平流速度场，以及仅可积的初值数据。我们证明，在非结构正则网格上，该迎风格式生成的近似解向连续模型唯一解收敛的速率至少为一阶。数值误差通过对数形式的Kantorovich-Rubinstein（坎托罗维奇-鲁宾斯坦）距离进行估计，从而给出了弱收敛速率的一个界。
英文摘要：
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.
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