The theta-method for ODEs with divergence-free Sobolev velocity fields converges to the regular Lagrangian flow at rate 1/|log h| in L1.
Ambrosio : Transport equation and Cauchy problem for BV vector fields
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A priori error estimates for the $\theta$-method for the flow of nonsmooth velocity fields
The theta-method for ODEs with divergence-free Sobolev velocity fields converges to the regular Lagrangian flow at rate 1/|log h| in L1.