UC San Diego

The accurate space-time discretization of the partial differential equations (PDEs) governing the dynamic behavior of complex physical phenomena is a core challenge in the field of Computational Fluid Dynamics, and in the simulation of turbulence in particular. However, it appears that over the last 30 years a disproportionate amount of attention has been addressed toward the improvement of spatial discretization techniques, while temporal discretization has relied, in most cases, on old consolidated approaches. Large Eddy Simulation (LES) and Direct Numerical Simulation (DNS) of the incompressible Navier-Stokes Equation (NSE) today often use a mixed implicit/explicit (IMEX) time integration approach developed in the mid 1980s, which combines the second-order implicit Crank-Nicolson (CN) method for the integration of the linear stiff terms and a third-order explicit low-storage Runge-Kutta-Wray (RKW3) method for the nonlinear terms. This hybrid approach, dubbed CN/RKW3, guarantees overall second-order accuracy for the time integration, while allowing an efficient storage implementation.

Our work focuses on the development of new mixed implicit/explicit time integration schemes of the Runge-Kutta type for the simulation of high-dimensional stiff ODEs, with particular attention to the simulation of the NSE. Compared with the venerable CN/RKW3 method, our numerical schemes have better accuracy, improved stability properties, and require the same or slightly increased storage.

We have also developed new relaxation schemes for the iterative solution of linear and, with some modification, nonlinear systems arising from the discretization of PDEs. These schemes prove especially advantageous when applied as the smoothing step in the multigrid solution of elliptic PDEs over stretched grids. A noteworthy application is the iterative solution of the pressure Poisson equation arising when imposing the diverge-free constraint during the simulation of the incompressible NSE using a fractional step method. Compared with the standard approach, our schemes require significantly less computation, while providing comparable converge rates.

We then discuss the implementation of an Ensemble Kalman Filter (EnKF) for the short-term prediction of ocean waves. The approach leverages one of our low-storage IMEX Runge-Kutta schemes for the highly-resolved simulation of the nonlinear equations used to describe wave propagation. We found that, using EnKF for data assimilation of wave measurement data, it is possible to perform accurate wave forecasting up to thirty seconds into the future, provided a sufficient number of ensemble members is employed.

Finally, we introduce a new direct multiple shooting algorithm for Nonlinear Model Predictive Control (NMPC). The new approach allows analytic calculation of the discretized trajectories and associated gradients, which are required when solving the nonlinear programming problem arising within the NMPC formulation. For the discretization of the trajectories, two solutions are proposed: one based on a Runge-Kutta discretization of the continuous-time model, and one leveraging a nonlinear discrete-time model based on Taylor-Lie algebra. This algorithm is then applied to the optimization of the power take-off of a point-absorber wave energy converter (WEC). Results have shown that NMPC improves the WEC power take-off with respect to linear MPC, since the nonlinear viscous forces affecting WEC dynamics are better accounted for. Moreover, the nonlinear formulation also allows the investigation of more complex configurations, such as one-way power flow.