Combining a least-squares approximate Jacobian with an analytical model to couple a flow solver with free surface position updates

This paper presents a new quasi-Newton method suitable for systems that can be solved with a black-box solver for which a cheap surrogate model is available. In order to have fast convergence, the approximate Jacobian consists of two different contribution: a full rank surrogate model of the system is combined with a low rank least-squares model based on known input-output pairs of the system. It is then shown how this method can be used to solve 2D steady free surface flows with a black-box flow solver. The inviscid flow over a ramp is calculated for supercritical and subcritical conditions. For both simulations the quasi-Newton iterations converge exponentially and the results match the analytical predictions accurately.