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A semi-implicit stochastic multiscale method for radiative heat transfer problem

Shan Zhang, Yajun Wang, Xiaofei Guan

TL;DR

This work tackles time-dependent radiative heat transfer in composite materials with random microstructure by formulating a semi-implicit stochastic multiscale method. It combines a predictor-corrected time discretization for the nonlinear radiation term, a finite-dimensional representation of additive noise, and CEM-GMsFEM-based multiscale basis functions to reduce spatial dimensionality while preserving accuracy. The authors prove convergence, deriving error bounds that separate noise truncation from multiscale discretization, and show optimal rates under high-contrast coefficients. Numerical experiments on periodic and non-periodic microstructures demonstrate strong accuracy and efficiency, confirming the method’s robustness and applicability to complex stochastic multiscale radiative transfer problems.

Abstract

In this paper, we propose and analyze a new semi-implicit stochastic multiscale method for the radiative heat transfer problem with additive noise fluctuation in composite materials. In the proposed method, the strong nonlinearity term induced by heat radiation is first approximated, by a semi-implicit predictor-corrected numerical scheme, for each fixed time step, resulting in a spatially random multiscale heat transfer equation. Then, the infinite-dimensional stochastic processes are modeled and truncated using a complete orthogonal system, facilitating the reduction of the model's dimensionality in the random space. The resulting low-rank random multiscale heat transfer equation is approximated and computed by using efficient spatial basis functions based multiscale method. The main advantage of the proposed method is that it separates the computational difficulty caused by the spatial multiscale properties, the high-dimensional randomness and the strong nonlinearity of the solution, so they can be overcome separately using different strategies. The convergence analysis is carried out, and the optimal rate of convergence is also obtained for the proposed semi-implicit stochastic multiscale method. Numerical experiments on several test problems for composite materials with various microstructures are also presented to gauge the efficiency and accuracy of the proposed semi-implicit stochastic multiscale method.

A semi-implicit stochastic multiscale method for radiative heat transfer problem

TL;DR

This work tackles time-dependent radiative heat transfer in composite materials with random microstructure by formulating a semi-implicit stochastic multiscale method. It combines a predictor-corrected time discretization for the nonlinear radiation term, a finite-dimensional representation of additive noise, and CEM-GMsFEM-based multiscale basis functions to reduce spatial dimensionality while preserving accuracy. The authors prove convergence, deriving error bounds that separate noise truncation from multiscale discretization, and show optimal rates under high-contrast coefficients. Numerical experiments on periodic and non-periodic microstructures demonstrate strong accuracy and efficiency, confirming the method’s robustness and applicability to complex stochastic multiscale radiative transfer problems.

Abstract

In this paper, we propose and analyze a new semi-implicit stochastic multiscale method for the radiative heat transfer problem with additive noise fluctuation in composite materials. In the proposed method, the strong nonlinearity term induced by heat radiation is first approximated, by a semi-implicit predictor-corrected numerical scheme, for each fixed time step, resulting in a spatially random multiscale heat transfer equation. Then, the infinite-dimensional stochastic processes are modeled and truncated using a complete orthogonal system, facilitating the reduction of the model's dimensionality in the random space. The resulting low-rank random multiscale heat transfer equation is approximated and computed by using efficient spatial basis functions based multiscale method. The main advantage of the proposed method is that it separates the computational difficulty caused by the spatial multiscale properties, the high-dimensional randomness and the strong nonlinearity of the solution, so they can be overcome separately using different strategies. The convergence analysis is carried out, and the optimal rate of convergence is also obtained for the proposed semi-implicit stochastic multiscale method. Numerical experiments on several test problems for composite materials with various microstructures are also presented to gauge the efficiency and accuracy of the proposed semi-implicit stochastic multiscale method.
Paper Structure (13 sections, 9 theorems, 104 equations, 12 figures, 3 tables)

This paper contains 13 sections, 9 theorems, 104 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Multiscale analysis of stochastic radiative heat transfer problems
  3. Problem setup
  4. The representation of additive noise
  5. Formulae of stochastic CEM-GMsFEM
  6. Semi-implicit predictor-corrected numerical scheme with spatial discretization methods
  7. The convergence analysis
  8. Approximate truncated error estimate
  9. Error estimation for the multiscale method
  10. Numerical results
  11. Example 1
  12. Example 2
  13. Conclusions

Key Result

lemma thmcounterlemma

Let $\zeta(t)=\int_0^t e^{-(t-s)A}dW(s)-\int_0^t e^{-(t-s)A}dW_{n}(s)$, there hold where and are infinite column vectors of noise's coefficients. The seminorm is defined by where $Q_{s}$ represents the infinite matrix with entries $Q_{s}=((kl)^{s}q_{kl})_{k,l=1}^{\infty}$ for an integer $s$, and $q_{kl}$ is defined by Eq.(eq:qkl).

Figures (12)

  • Figure 1: An illustration of a sample path of Brownian motion using cumulative summation of increments
  • Figure 2: Truncated noise in orthogonal basis forms (\ref{['1noise']}) for various values of $n=4,8,16,32$.
  • Figure 3: Truncated noise in Fourier forms (\ref{['appnoise']}) with decay coefficients $\gamma_{k}=\frac{1}{k^{3/2}}$ for various values of $n=4,8,16,32$.
  • Figure 4: Truncated noise in Fourier forms (\ref{['appnoise']}) with decay coefficients $\gamma_{k}=\frac{1}{2^{k}}$ for various values of $n=4,8,16,32$.
  • Figure 5: The fine grid, coarse grid $K_{i}$, oversampling domain $K_{i,1}$, and neighborhood $\omega_{i}$ of the nodes $x_{i}$.
  • ...and 7 more figures

Theorems & Definitions (18)

  • definition thmcounterdefinition: $\phi_{j}^{(i)}$-orthogonality
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • theorem 1
  • proof
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • remark thmcounterremark
  • ...and 8 more