Ensemble Time-Stepping Algorithm for the Convection-Diffusion Equation with Random Diffusivity

In this paper, we develop two ensemble time-stepping algorithms to solve the convection-diffusion equation with random diffusion coefficients, forcing terms and initial conditions based on the pseudo-spectral stochastic collocation method. The key step of the pseudo-spectral stochastic collocation m...

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Veröffentlicht in:Journal of scientific computing 2019-05, Vol.79 (2), p.1271-1293
Hauptverfasser: Li, Ning, Fiordilino, Joseph, Feng, Xinlong
Format: Artikel
Sprache:eng
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Zusammenfassung:In this paper, we develop two ensemble time-stepping algorithms to solve the convection-diffusion equation with random diffusion coefficients, forcing terms and initial conditions based on the pseudo-spectral stochastic collocation method. The key step of the pseudo-spectral stochastic collocation method is to solve a number of deterministic problems derived from the original stochastic convection-diffusion equation. In general, a common way to solve the set of deterministic problems is by using the backward differentiation formula, which requires us to store the coefficient matrix and right-hand-side vector multiple times, and solve them one by one. However, the proposed algorithm only need to solve a single linear system with one shared coefficient matrix and multiple right-hand-side vectors, reducing both storage required and computational cost of the solution process. The stability and error analysis of the first- and second-order ensemble time-stepping algorithms are provided. Several numerical experiments are presented to confirm the theoretical analyses and verify the feasibility and effectiveness of the proposed method.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-018-0890-8