A numerical B-spline Galerkin method with proper generalized decomposition for reduced order modeling of partial differential equations

Abstract

A numerical study is presented on reduced order modeling of low and high-dimensional partial differential equations using a new B-spline Galerkin proper generalized decomposition (PGD) method. Galerkin-PGD schemes based on B-splines are numerically investigated for various benchmark problems, including the Poisson equation, linear/nonlinear advection-diffusion equations, and the Navier–Stokes equations. The treatment of inhomogeneous boundary conditions, non-separable terms and nonlinearities is numerically investigated. Numerical errors for high-fidelity B-spline solutions and reduced order solutions are provided to demonstrate the effectiveness and accuracy of the proposed method. Computational costs are reported to illustrate the efficiency of the developed approaches.