Real-time reduced order modeling of deterministic partial differential equations using time dependent basis

A new methodology using time dependent basis is presented for solving deterministic partial differential equations. In this method, a multi-dimensional variable is represented by a set of time-dependent orthonormal modes in each dimension and a core tensor. The best update for the modes at every time step is found as the solution evolves. To achieve lower computational cost for non-polynomial non-linearity in the equation an adaptive sparse interpolation algorithm is implemented. Thus a computational speedup is achieved using the discrete empirical interpolation method (DEIM). This method is implemented to solve the incompressible and compressible Navier-Stokes equations. The error convergence properties of the method for different reduction orders are compared. We also demonstrate the development of the modes according to the evolution of the flow physics. Due to the low rank representation of the solution at every time instant, the method also provides advantages in data storage for a large number of time steps. This advantage is notably evident in higher dimensions (d>2).