Convergence Rates for Finite Volume Scheme of Stochastic Heat Equation with Multiplicative Noise

Core Concepts

The authors provide convergence rates for the finite volume scheme of the stochastic heat equation with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions.

Abstract

The content discusses the convergence rates for the finite volume scheme of the stochastic heat equation (SHE) with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions.
Key highlights:

The authors consider the case of two and three spatial dimensions and provide error estimates for the L2-norm of the space-time discretization of the SHE and the variational solution.
The error estimates are of order O(τ^(1/2) + h + hτ^(-1/2)), where τ represents the time step and h the spatial parameter.
The main idea is to compare the exact solution of the SHE with the solution of the semi-implicit Euler scheme, then the solution of the semi-implicit Euler scheme with its parabolic projection, and finally the parabolic projection with the solution of the finite volume scheme.
Regularity assumptions on the initial condition u0 (H2-regularity in space) and the function g in the stochastic Itô integral (smoothness) are needed to obtain the convergence rates.
The stochastic nature of the problem creates worse convergence rates compared to the deterministic case.

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Convergence rates for the finite volume scheme of the stochastic heat equation

by Niklas Sapou... at arxiv.org 04-09-2024

https://arxiv.org/pdf/2404.05655.pdf

Deeper Inquiries

What are the potential applications of the finite volume scheme for the stochastic heat equation beyond the theoretical analysis presented in this work

The finite volume scheme for the stochastic heat equation presented in the context above has various potential applications beyond theoretical analysis. One significant application is in computational modeling of physical systems with stochastic heat transfer phenomena. This scheme can be utilized to simulate heat conduction in materials with random fluctuations in thermal properties or boundary conditions. Industries such as materials science, engineering, and environmental science could benefit from accurate simulations of heat distribution considering stochastic influences. Additionally, the scheme could be applied in the study of random diffusion processes in complex systems where heat plays a crucial role, such as in biological systems or geophysical processes. Overall, the scheme provides a versatile tool for understanding and predicting heat dynamics in stochastic environments.

How could the convergence rates be improved, for example, by considering additional regularity assumptions or alternative numerical schemes

To improve the convergence rates of the finite volume scheme for the stochastic heat equation, several strategies can be considered. One approach is to impose stronger regularity assumptions on the initial data and the diffusive term. By assuming higher regularity of these components, the error estimates could potentially be reduced, leading to faster convergence rates. Additionally, exploring alternative numerical schemes with better stability properties and higher order accuracy could enhance the convergence rates. Techniques like adaptive mesh refinement, implicit time integration methods, or higher-order spatial discretization schemes could be investigated to improve the overall efficiency and accuracy of the scheme. By combining advanced numerical methods with appropriate regularity assumptions, the convergence rates of the scheme can be significantly enhanced.

What are the implications of the worse convergence rates in the stochastic case compared to the deterministic case, and how could this be addressed in future research

The worse convergence rates in the stochastic case compared to the deterministic case pose challenges in accurately modeling and simulating stochastic heat equations. This discrepancy is primarily due to the inherent randomness in the system, which introduces additional complexities and uncertainties that affect the convergence behavior of numerical schemes. To address this issue in future research, advanced stochastic analysis techniques and probabilistic methods could be employed to better understand the behavior of stochastic heat equations. Developing novel numerical algorithms specifically tailored for stochastic systems, such as stochastic Galerkin methods or probabilistic collocation techniques, could help mitigate the impact of the slower convergence rates. Furthermore, exploring hybrid approaches that combine deterministic and stochastic methods could offer a promising direction to improve convergence rates and computational efficiency in stochastic heat equation simulations.