The paper introduces a finite difference scheme for approximating weak L2 solutions of the stochastic transport equation:
du + V · ∇u dt + σ∇u ◦ dW = 0
where V is the velocity field and σ is the noise coefficient. The authors allow for low-regularity velocity fields V, only requiring V ∈ L2 ∩ L∞ with ∂x1V 1, ..., ∂xdV d ∈ Lp for some p > d, in contrast to the typical assumption of div V ∈ L∞.
The key aspects of the analysis are:
The discretization of the noise term σ∇u ◦ dW is designed to preserve the hyperbolic character of the equation, leading to a crucial structural property (1.3) that enables L2 stability.
A discrete duality argument is developed, building on a detailed analysis of a class of backward parabolic difference schemes. This allows the authors to obtain L2 stability estimates that are robust to the low regularity of the velocity field V.
Consistency and convergence of the scheme are established, showing that the numerical approximations converge to the unique weak L2 solution of the stochastic transport equation.
The paper represents one of the first attempts to leverage noise-based regularization to construct consistent and stable numerical schemes for stochastic transport equations with low-regularity coefficients.
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by Ulrik S. Fjo... at arxiv.org 04-22-2024
https://arxiv.org/pdf/2309.02208.pdfDeeper Inquiries