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Convergent Finite Difference Schemes for Stochastic Transport Equations with Low-Regularity Velocity Fields


Core Concepts
This paper presents and analyzes convergent finite difference schemes for approximating weak L2 solutions of stochastic transport equations with low-regularity velocity fields. The schemes are shown to be stable and convergent under less restrictive conditions than those required for deterministic transport equations.
Abstract
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.
Stats
The following sentences contain key metrics or figures: The velocity field V lies in L2 ∩ L∞(Rd) with ∂x1V 1, ..., ∂xdV d ∈ Lp(Rd) for some p > d. The noise coefficient σ lies in the Sobolev space W 3,∞(Rd) and satisfies σ(x) ≥ σ0 for all x ∈ Rd for some number σ0 > 0. The initial data u0 lies in L2(Rd).
Quotes
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Deeper Inquiries

How can the proposed numerical scheme be extended to handle time-dependent velocity fields V(t,x)

The extension of the proposed numerical scheme to handle time-dependent velocity fields V(t,x) involves incorporating the time variable into the discretization process. In the current scheme, the velocity field V is treated as a constant with respect to time. To adapt to time-dependent velocities, the discrete scheme needs to account for the evolution of V over time. This can be achieved by introducing a time discretization method, such as an implicit scheme, to update the velocity field at each time step. The discretization of the time-dependent velocity field would involve updating the grid values of V at each time iteration, ensuring that the scheme accurately captures the dynamics of the changing velocity field over time. By incorporating the time variable into the discretization process, the numerical scheme can effectively handle time-dependent velocity fields in the stochastic transport equation.

What are the potential applications of the stochastic transport equation and the developed numerical methods in fields such as fluid mechanics or climate modeling

The stochastic transport equation and the developed numerical methods have various potential applications in fields such as fluid mechanics and climate modeling. In fluid mechanics, the stochastic transport equation can be used to model the transport of particles or substances in turbulent flows where the velocity field exhibits randomness or uncertainty. The numerical methods developed for this equation provide a framework for simulating and analyzing the behavior of these stochastic transport processes, offering insights into the dispersion, mixing, and diffusion of particles in complex flow environments. In climate modeling, the stochastic transport equation can be applied to study the dispersion of pollutants, aerosols, or greenhouse gases in the atmosphere. By incorporating stochasticity into the transport processes, the model can account for the inherent variability and uncertainties in atmospheric dynamics, leading to more realistic simulations of pollutant dispersion and climate change impacts. The numerical methods developed for the stochastic transport equation enable researchers to simulate and analyze the complex interactions between atmospheric transport processes and environmental factors, contributing to improved understanding and prediction of climate phenomena.

Can the underlying principles of the analysis be applied to other types of stochastic partial differential equations beyond the stochastic transport equation considered in this work

The underlying principles of the analysis presented in the context of the stochastic transport equation can be extended to other types of stochastic partial differential equations (SPDEs) beyond the specific equation considered in this work. The key techniques of discretization, stability analysis, and convergence proofs can be applied to a wide range of SPDEs with stochastic forcing terms, variable coefficients, or irregular data. By adapting the numerical methods and analytical approaches developed for the stochastic transport equation, researchers can address various SPDEs encountered in diverse fields such as mathematical biology, finance, geophysics, and materials science. The general framework of constructing finite difference schemes, establishing stability criteria, and proving convergence results can be tailored to suit the specific characteristics and requirements of different types of SPDEs. By leveraging the insights and methodologies from the analysis of the stochastic transport equation, researchers can develop robust numerical schemes and analytical tools for studying and solving a broad class of stochastic partial differential equations in various scientific and engineering applications.
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