A direct application of the unsteady output adjoint method for error estimation is not possible for chaotic flows due to their inherent sensitivity to the initial conditions and the exponential growth of their corresponding adjoint solutions. A method that has proven to provide accurate and usable output adjoints for chaotic flows is the least squares shadowing method (LSS), which was originally developed for sensitivity analysis. However, this approach has been shown to be costly, especially for larger simulations. Another method, the approximate time-windowing approach, has been shown to be cheaper to implement and execute. However, this method produces results that are not as accurate as that of LSS. In this paper, we choose to concentrate on reducing the computational cost associated with LSS by using reduced-order modeling (ROM) and hyper-reduced-order modeling (HROM). We first present a study on how accurate ROM and HROM can be for a chaotic system. We then introduce a combined reduced-order model, least squares shadowing method (HROM-LSS) for approximating output adjoints more accurately and more economically compared to previous approaches. Lastly, we present preliminary HROM-LSS results for the 1D chaotic Kuramoto-Sivashinsky (KS) problem.

This paper presents two methods for estimating the effect of numerical discretization error on statistical output accuracy in chaotic unsteady flow simulations: (1) an extension of recent advances in least-squares shadowing sensitivity calculations (LSS), and (2) an approximate time-windowing approach with individual adjoint solutions on each time window. Both methods rely on output adjoints, a direct application of which is not possible for chaotic systems, in which sensitivities to initial conditions and single-point discretization errors grow exponentially. This paper shows results for two prototypical chaotic systems: the Lorenz oscillator and the modified ergodic Kuramoto-Sivashinksy partial differential equation (MEKS). In addition it presents results for a low-Reynolds number Navier-Stokes flow to demonstrate the effectivity of the error estimates. Preliminary adaptive results are also included, in which spatially-localized forms of the error estimates drive static mesh adaptation to reduce errors in statistical outputs.

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