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Long time numerical stability of implicit schemes for stochastic heat equations

Xiaochen Yang, Yaozhong Hu

TL;DR

The paper addresses long-time stability of stochastic heat equations on bounded domains driven by spatially colored noise, contrasting with intermittency observed on unbounded domains. It employs an eigenfunction expansion of the Dirichlet Laplacian to recast the SPDE as an infinite-dimensional SDE and derives a direct mean-square stability condition $2(λ_1+β_0) - β_1^2 κ > 0$, where $κ = \sup_{ξ∈O} q(ξ,ξ)$. A spectral Galerkin approximation preserves stability under $2(λ_1+β_0) - β_1^2 \tilde{κ}_1 > 0$ with $\tilde{κ}_1 = \sup_N ||q_N||_2$ and exhibits convergence with error $\|u(t) - U_N(t)\|_2^2 ≤ C(λ_N^{-1+γ} + ρ(M))$, where ρ(M) → 0 as M → ∞. The full discretization using implicit Euler is shown to be mean-square stable under the same spectral condition, and numerical experiments corroborate the theory, indicating practical stability and reliable spectral approximations for bounded domains.

Abstract

This paper studies the long time stability of both stochastic heat equations on a bounded domain driven by a correlated noise and their approximations. It is popular for researchers to prove the intermittency of the solution which means that the moments of solution to stochastic heat equation usually grow exponentially to infinite and this hints that the solution to stochastic heat equation is generally not stable in long time. However, quite surprisingly in this paper we show that when the domain is bounded and when the noise is not singular in spatial variables, the system can be long time stable and we also prove that we can approximate the solution by its finite dimensional spectral approximation which is also long time stable. The idea is to use eigenfunction expansion of the Laplacian on bounded domain. We also present numerical experiments which are consistent with our theoretical results.

Long time numerical stability of implicit schemes for stochastic heat equations

TL;DR

The paper addresses long-time stability of stochastic heat equations on bounded domains driven by spatially colored noise, contrasting with intermittency observed on unbounded domains. It employs an eigenfunction expansion of the Dirichlet Laplacian to recast the SPDE as an infinite-dimensional SDE and derives a direct mean-square stability condition , where . A spectral Galerkin approximation preserves stability under with and exhibits convergence with error , where ρ(M) → 0 as M → ∞. The full discretization using implicit Euler is shown to be mean-square stable under the same spectral condition, and numerical experiments corroborate the theory, indicating practical stability and reliable spectral approximations for bounded domains.

Abstract

This paper studies the long time stability of both stochastic heat equations on a bounded domain driven by a correlated noise and their approximations. It is popular for researchers to prove the intermittency of the solution which means that the moments of solution to stochastic heat equation usually grow exponentially to infinite and this hints that the solution to stochastic heat equation is generally not stable in long time. However, quite surprisingly in this paper we show that when the domain is bounded and when the noise is not singular in spatial variables, the system can be long time stable and we also prove that we can approximate the solution by its finite dimensional spectral approximation which is also long time stable. The idea is to use eigenfunction expansion of the Laplacian on bounded domain. We also present numerical experiments which are consistent with our theoretical results.
Paper Structure (7 sections, 9 theorems, 119 equations, 4 figures)

This paper contains 7 sections, 9 theorems, 119 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Long time stability of the solution
  3. Spectral Approximation
  4. Full discretization
  5. Other numerical method
  6. Numerical experiments
  7. Conclusions

Key Result

Theorem 2.1

Let $\kappa$ be given by e.2.10. If then the system Heatequation is stable in the mean square sense, namely, $\lim_{t\to \infty} \||u(t)\||_2=0$.

Figures (4)

  • Figure 1: The mean-square curves of equation (\ref{['Heatexampleequation']}).
  • Figure 2: Mean-square stability region of the numerical solutions and the analytical solutions.
  • Figure 3: Mean-square curve of numerical solutions with $\beta_0=1, \beta_1=1$.
  • Figure 4: Mean-square curve of numerical solutions with different numerical method.

Theorems & Definitions (22)

  • Theorem 2.1
  • proof
  • Remark 2.2
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Lemma 3.5
  • ...and 12 more