Sensitivity analysis methods are important tools for research and design with computational models like CFD. Traditional sensitivity analysis methods are unable to compute useful gradient information for long time averaged quantities in chaotic dynamical systems, such as high fidelity simulations of turbulent fluid flows. The Least Squares Shadowing (LSS) method has been used to compute useful gradient information for a number of chaotic systems, including a simulation of homogeneous isotropic turbulence. However, some LSS gradient calculations for the Kuramoto-Sivshinsky (K-S) equation and the Lorenz 96 system have a systematic error due to breaks in the assumption of ergodicity. Since these systems have similar characteristics to turbulent fluid flows, this ergodicity breaking error must be minimized. This paper proposes a new approach using LSS, Multiple Shooting Shadowing (MSS), which uses the multiple shooting implementation of LSS to reduce the size of the ergodicity breaking error by not running the multiple shooting algorithm to full convergence. This way, gradients are computed from an ensemble of solutions, rather than the shadow direction alone, making the method more robust to the ergodicity breaking error. In this paper, MSS is demonstrated for the K-S equation and it is found that MSS cannot fix the systematic error of LSS when the system has a wide range of chaotic time scales.
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