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Data-driven eigensolution analysis based on a spatio-temporal Koopman decomposition, with applications to high-order methods

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Abstract

We propose a data-driven method to perform eigensolution analyses and quantify numerical errors in a non-intrusive manner. In classic eigensolution analysis methods, explicit matrices need to be constructed, whilst in our approach only solution snapshots from numerical simulations are required to quantify the numerical errors (dispersion and diffusion) in time and/or space. This new approach is based on a recent data-driven method: the Spatio-Temporal Koopman Decomposition (STKD), that approximates spatio-temporal data as a linear combination of standing or travelling waves growing or decaying exponentially in time and/or space. We validate our approach with classic matrix-based approaches, where accurate predictions of the dispersion-dissipation behaviour for both temporal and spatial eigensolution analyses are reported.

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... For eigensolution analysis of high-order methods, more details can be found in [77,80]. It is also worth noting the recent work on data-driven eigensolution analysis [81], where the dispersion-dissipation behaviors are extracted purely from simulation data in a non-intrusive manner. Since IBM source term is only introduced to the solid region, and the global discretization matrix in Equation (A.7) is used, the physical mode that recovers k * = k as k → 0 needs to be extracted properly [78,56]. ...
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... A machine learning model involves a mapping from data to data, and the generalization capability of the model depends on the range of training sample data available, and so it is easy to obtain ideal fitting results in the envelope range of the training data set (here 1.5-3.0 kPa), whereas for extrapolation, i.e., outside the envelope range of the training set (here 4.0 kPa), the prediction accuracy will be reduced [31][32][33][34][35]. ...
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... To investigate the evolution in space, spatial eigensolution analysis [43,44] has recently been proposed for high-order schemes with inflow-outflow boundary conditions. The dispersion-dissipation behaviour can also be obtained purely from simulation data, as shown in a recent work by the authors [45]. Detailed explanation of the eigensolution analysis is given in Appendix A. To assess the jump penalty stabilisation, the limit of a very large Péclet number is considered, such that viscous diffusion is negligible. ...
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Jump penalty stabilisation techniques have been recently proposed for continuous and discontinuous high order Galerkin schemes [1,2,3]. The stabilisation relies on the gradient or solution discontinuity at element interfaces to incorporate localised numerical diffusion in the numerical scheme. This diffusion acts as an implicit subgrid model and stablises under-resolved turbulent simulations. This paper investigates the effect of jump penalty stabilisation methods (penalising gradient or solution) for stabilisation and improvement of high-order discontinuous Galerkin schemes in turbulent regime. We analyse these schemes using an eigensolution analysis, a 1D non-linear Burgers equation (mimicking a turbulent cascade) and 3D turbulent Navier-Stokes simulations (Taylor-Green Vortex problem). We show that the two jump penalty stabilisation techniques can stabilise under-resolved simulations thanks to the improved dispersion-dissipation characteristics (when compared to non-penalised schemes) and provide accurate results for turbulent flows. The numerical results indicate that the proposed jump penalty stabilise under-resolved simulations and improve the simulations, when compared to the original unpenalised scheme and to classic explicit subgrid models (Smagorisnky and Vreman).
... To investigate the evolution in space, spatial eigensolution analysis [18,19] has been proposed recently for high-order schemes with inflow-outflow boundary conditions. Both spatial and temporal eigensolution analyses have been extended in a non-intrusive and data-driven manner to evaluate dispersion-dissipation behavior only from simulation data [20]. Another alternative perspective for the dissipation error is to look at the non-modal behavior characterized by the short-term dissipation of a numerical scheme [21], which has been shown to obtain useful insights for the hybridized Discontinuous Galerkin (DG) scheme. ...
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