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On a finite-volume approximation of a diffusion-convection equation with a multiplicative stochastic force

Caroline Bauzet, Kerstin Schmitz, Aleksandra Zimmermann

TL;DR

The paper analyzes a diffusion-convection SPDE with multiplicative noise on a bounded domain under Neumann conditions and proposes a semi-implicit finite-volume scheme using upwind for the convection and TPFA for diffusion. By combining time-discretization techniques for SPDEs with deterministic spatial discretization tools, it achieves strong convergence of both right- and left-time approximations to the unique variational solution, without relying on stochastic compactness arguments. The convergence is established in $L^p(0,T;L^2(\Omega;L^2(\Lambda)))$ for all finite $p\ge1$ in dimensions $d=2$ and $3$, with $g(u)$ and $\beta(u)$ identified in the limit, and the approach handles the convection term and multiplicative noise in a robust, implementable framework. This work advances numerical analysis for SPDEs by providing a structure-preserving, convergence-guaranteed scheme that can be extended to broader nonlinear flux settings and higher dimensions, enhancing reliability of simulations in stochastic diffusion-convection models.

Abstract

We address an original approach for the convergence analysis of a finite-volume scheme for the approximation of a stochastic diffusion-convection equation with multiplicative noise in a bounded domain of $\mathbb{R}^d$ (with $d=2$ or $3$) and with homogeneous Neumann boundary conditions. The idea behind our approach is to avoid using the stochastic compactness method. We study a numerical scheme that is semi-implicit in time and in which the convection and the diffusion terms are respectively approximated by means of an upwind scheme and the so called two-point flux approximation scheme (TPFA). By adapting well-known methods for the time discretization of stochastic PDEs and combining them with deterministic techniques applied to spatial discretization, we show strong convergence of our scheme towards the unique variational solution of the continuous problem in $L^p(0,T;L^2(Ω;L^2(Λ)))$, for any finite $p\geq 1$.

On a finite-volume approximation of a diffusion-convection equation with a multiplicative stochastic force

TL;DR

The paper analyzes a diffusion-convection SPDE with multiplicative noise on a bounded domain under Neumann conditions and proposes a semi-implicit finite-volume scheme using upwind for the convection and TPFA for diffusion. By combining time-discretization techniques for SPDEs with deterministic spatial discretization tools, it achieves strong convergence of both right- and left-time approximations to the unique variational solution, without relying on stochastic compactness arguments. The convergence is established in for all finite in dimensions and , with and identified in the limit, and the approach handles the convection term and multiplicative noise in a robust, implementable framework. This work advances numerical analysis for SPDEs by providing a structure-preserving, convergence-guaranteed scheme that can be extended to broader nonlinear flux settings and higher dimensions, enhancing reliability of simulations in stochastic diffusion-convection models.

Abstract

We address an original approach for the convergence analysis of a finite-volume scheme for the approximation of a stochastic diffusion-convection equation with multiplicative noise in a bounded domain of (with or ) and with homogeneous Neumann boundary conditions. The idea behind our approach is to avoid using the stochastic compactness method. We study a numerical scheme that is semi-implicit in time and in which the convection and the diffusion terms are respectively approximated by means of an upwind scheme and the so called two-point flux approximation scheme (TPFA). By adapting well-known methods for the time discretization of stochastic PDEs and combining them with deterministic techniques applied to spatial discretization, we show strong convergence of our scheme towards the unique variational solution of the continuous problem in , for any finite .
Paper Structure (14 sections, 15 theorems, 169 equations, 1 figure)

This paper contains 14 sections, 15 theorems, 169 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations
  3. Concept of solution and main result
  4. State of the art
  5. Originality of the study and outline of the paper
  6. The finite-volume framework
  7. Admissible finite-volume meshes and notations
  8. Discrete unknowns and piecewise constant functions
  9. Discrete norms and weak gradient
  10. The finite-volume scheme
  11. Stability estimates
  12. Convergence of the finite-volume scheme
  13. Weak convergence of finite-volume approximations
  14. Identification of weak limits coming from the non-linear terms

Key Result

Theorem 1.4

Assume that hypotheses $\mathscr{A}_1$ to $\mathscr{A}_4$ hold. Let $(\mathcal{T}_m)_{m\in \mathbb{N}}$ be a sequence of admissible finite-volume meshes of $\Lambda$ in the sense of Definition defmesh such that the mesh size $h_m$ tends to $0$ and let $(N_m)_{m\in \mathbb{N}}\subset \mathbb{N}^{\sta

Figures (1)

  • Figure 1: Notations of the mesh $\mathcal{T}$ associated with $\Lambda$ for $d=2$

Theorems & Definitions (42)

  • Remark 1.1
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4: Discrete $L^2$-norm
  • Definition 2.5: Weak gradient
  • ...and 32 more