<p>We study the generalized multi-parameter stochastic time-fractional damped wave equation, which arises in modeling wave propagation in complex media with memory and random perturbations. Its numerical analysis is challenging due to the low temporal regularity induced by stochastic forcing and the high computational costs associated with nonlocal fractional dynamics. To address these issues, we construct an approximate equation via the Wong–Zakai approximation and employ frequency-domain analysis to avoid the direct treatment of generalized Mittag–Leffler functions. Based on this framework, we design a spectral Galerkin discretization in space combined with exponential Euler and exponential trapezoidal time-stepping schemes, and establish parameter-dependent strong convergence rates that, in certain regimes, surpass existing results on temporal accuracy. To enhance computational efficiency, we propose a Fast Exponential Integrator Method (FEIM), which integrates the sum-of-exponentials method with fast time-stepping techniques and reduces complexity from <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {O}(M^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mi>M</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {O}(M\log M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo>log</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and storage from <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {O}(M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {O}(\log M)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <mo>log</mo> <mi>M</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Numerical experiments confirm the theoretical findings and demonstrate the capability of FEIM for efficient high-dimensional simulations.</p>

Fast exponential integrator method for stochastic time-fractional damped wave equations via Wong–Zakai approximation

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发表日期：

2026年7月10日

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Chunbin Li,
Yibo Wang,
Wanrong Cao

摘要

We study the generalized multi-parameter stochastic time-fractional damped wave equation, which arises in modeling wave propagation in complex media with memory and random perturbations. Its numerical analysis is challenging due to the low temporal regularity induced by stochastic forcing and the high computational costs associated with nonlocal fractional dynamics. To address these issues, we construct an approximate equation via the Wong–Zakai approximation and employ frequency-domain analysis to avoid the direct treatment of generalized Mittag–Leffler functions. Based on this framework, we design a spectral Galerkin discretization in space combined with exponential Euler and exponential trapezoidal time-stepping schemes, and establish parameter-dependent strong convergence rates that, in certain regimes, surpass existing results on temporal accuracy. To enhance computational efficiency, we propose a Fast Exponential Integrator Method (FEIM), which integrates the sum-of-exponentials method with fast time-stepping techniques and reduces complexity from $\mathcal {O}(M^2)$ $O (M^{2})$ to $\mathcal {O}(M\log M)$ $O (M log M)$ and storage from $\mathcal {O}(M)$ $O (M)$ to $\mathcal {O}(\log M)$ $O (log M)$ . Numerical experiments confirm the theoretical findings and demonstrate the capability of FEIM for efficient high-dimensional simulations.