RONS: Reduced-order nonlinear solutions for PDEs with conserved quantities

Reduced-order models where the solution is assumed as a linear combination of prescribed modes are rooted in a well-developed theory. However, more general models where the reduced solutions depend nonlinearly on time-dependent variables have thus far been derived in an ad hoc manner. Here, we introduce Reduced-order Nonlinear Solutions (RONS): a unified framework for deriving reduced-order models that depend nonlinearly on a set of time-dependent variables. The set of all possible reduced-order solutions are viewed as a manifold immersed in the function space of the PDE. The parameters are evolved such that the instantaneous discrepancy between reduced dynamics and the full PDE dynamics is minimized. In the special case of linear parameter dependence, our reduced equations coincide with the standard Galerkin projection. Furthermore, any number of conserved quantities of the PDE can readily be enforced in our framework. Since RONS does not assume an underlying variational formulation for the PDE, it is applicable to a broad class of problems, and in particular fluid dynamics. We demonstrate the efficacy of RONS on three fluid flows: an advection-diffusion equation, the propagation of surface waves, and vortex dynamics in ideal fluids.