Nonlinear Reduced-Order Solutions for PDEs With Conserved Quantities: Applications to Fluid Dynamics

Reduced-order Nonlinear Solutions (RONS) is a new framework for reduced-order modeling of PDEs where the solution depends nonlinearly on time-varying variables [1]. RONS views all possible reduced solutions as a manifold embedded in the function space of the PDE. The time-dependent variables are evolved so that the instantaneous error between true dynamics of the PDE and dynamics of the reduced model are minimized. Additionally, in the RONS framework, any number of conserved quantities of the PDE can be easily enforced. In this talk, we demonstrate the application of RONS on several canonical problems: an advection-diffusion equation, the nonlinear Schrödinger equation for water waves, a modified nonlinear Schrödinger equation, and vortex dynamics in ideal fluids. In all scenarios, RONS accurately approximates the true solutions of the underlying PDEs at a low computational cost.