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Probabilistic analysis of the upwind scheme for transport equations. (English) Zbl 1230.65008
The authors associate the upwind scheme for the $$d$$-dimensional transport problem $\begin{cases} \partial_{t}u(t,x)+\langle a(x),\nabla u(t,x)\rangle=0,\;(t,x)\in \mathbb{R}_{+}\times\mathbb{R}^{d},\\ u(0,x)=u^{0}(x),\;x\in\mathbb{R}^{d},\end{cases}$
with a Markov chain. They show that the error induced by the scheme is governed by the fluctuations of the Markov chain around characteristics of the considered problem. Using the probabilistic approach, the authors obtain some results concerning the order of convergence of the proposed scheme.

##### MSC:
 65C30 Numerical solutions to stochastic differential and integral equations 65C40 Numerical analysis or methods applied to Markov chains 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 35R60 PDEs with randomness, stochastic partial differential equations
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