From Time Series Expansion to Proper Generalized Decomposition via Graph-Theoretical Connection: Stabilized Simulation of Fluids Flow
In this paper, we employ graph theory to establish a connection between the Time Series Expansion (TSE) and Proper Generalized Decomposition (PGD) methods. Using the concept of a directed graph, we demonstrate how one can transition from the computation of space modes in the TSE--first illustrated for the diffusion equation--to those of space modes in PGD, in which an inhomogeneous Volterra-type convolution recurrence relation, weighted by time-dependent coefficients, appears. This recurrence relation is simplified through graph-based analysis into a compact form using a simple path traversal, reducing the computational complexity. Moreover, the compact formulation reveals a natural stabilization process in the computation of space modes, where stabilized coefficients are automatically derived and can be used in the Stabilized-TSE (STSE) framework. To explicitly construct these coefficients, we consider a Simplified PGD (SPGD) formulation in which the time modes are chosen to be the time polynomial basis $t^n$. This choice yields a one-level Volterra-type recurrence relation that is similarly simplified using a simple path representation, demonstrating a connection in the computation of space modes from TSE, through STSE and SPGD, to PGD. This graph-based connection is exhibited in the case of inviscid flow to check how crucial the addition of an artificial diffusion is in stabilizing the recurrence formula of TSE. Finally, we extend the approach to the incompressible, dimensionless Navier-Stokes (NS) equations and build stabilization coefficients that depend on the Reynolds number Re, the space mode rank, and the simulation time step. Both the STSE and SPGD approaches are tested to simulate the wake behind a bluff body at Re = 5 000.