Targeted ENO (TENO) has been proposed to overcome the shortcomings of WENO schemes, namely excessive dissipation of lower-order upwind-biased and degenerated schemes, and limited robustness of central-biased schemes. TENO offers a set of free parameters to shape the inherent effective local dissipation and dispersion. In the original formulation of TENO, these free parameters have been adjusting by means of the approximate dissipation-dispersion relation. Hence, the TENO formulation may be superior in this aspect, yet, it does not necessarily outperform other schemes in flows involving non-linear interaction of a broad range of scales. Data-driven methods enable optimizing these free parameters instead of adjusting them. In this work, we demonstrate the application of an iterative Bayesian optimization approach on designing fifth-order TENO (TENO5) schemes. Exploiting that Bayesian optimization efficiently and robustly finds an optimum of an expensive function with a low number of trials, we construct specific TENO5-schemes for compressible flows with gas dynamic discontinuities as well as for implicit large eddy simulation (ILES). For the former, we measure the error between under-resolved simulations of the Sod shock tube and its analytical solution for automatically generated TENO5 formulations as the objective. For the latter, under-resolved inviscid Taylor-Green vortex flows are evolved to their turbulent state, in which their kinetic energy spectrum in the inertial subrange is compared to the theoretical Kolmogorov-scaling solution to formulate its objective. We show that these two TENO5 formulations perform superior to the original formulation of TENO5 relevant to the specific types of flows. Also, a variety of benchmark test flows show that both specific TENO5 formulations outperform the original one in terms of phase speed, shock-preservation, as well as physical consistency of fluid-dynamic instabilities and turbulent flows.
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