A fast method for computing nonlinear reduced-order models with conserved quantities

Reduced-order nonlinear solutions (RONS) was developed recently as a powerful method for reduced-order modeling of PDEs. RONS significantly broadens the scope of reduced-order modeling compared to previous linear methods, e.g., Galerkin-type projections. RONS also allows for enforcing conserved quantities of the PDE in the reduced model. In this talk, I will discuss a fast and accurate method for forming the RONS equations. This method exploits the structure of the metric tensor involved in RONS to reduce the computational cost by several orders of magnitude. This speed up allows us to go beyond reduced-order modeling and use RONS for accurate numerical simulation of PDEs. I demonstrate the application of the algorithm on several examples including vortex dynamics in ideal fluids (Euler equation) and turbulence (Navier-Stokes equation). In particular, I will discuss the effect of enforcing conserved quantities on energy and enstrophy cascades.