Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

From Time Series Expansion to Proper Generalized Decomposition via Graph-Theoretical Connection: Stabilized Simulation of Fluids Flow

Ahmad Deeb, Vladimir Parezanovic, Denys Dutykh

Abstract

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.

From Time Series Expansion to Proper Generalized Decomposition via Graph-Theoretical Connection: Stabilized Simulation of Fluids Flow

Abstract

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 . 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.

Paper Structure

This paper contains 21 sections, 2 theorems, 100 equations, 14 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Graph theory background
  3. Connection between PGD and TSE for the diffusion problem
  4. Finding $\mathrm{u}_k$
  5. Finding $\mathcal{T}_n$
  6. Finding $\mathcal{X}_n$
  7. From PGD to TSE
  8. SPGD
  9. Summary
  10. Numerical performance and computational efficiency
  11. Application to Inviscid Flow
  12. Classical PGD method
  13. From TSE to SPGD
  14. Simulating the flow in a two-dimensional annulus.
  15. Application to Navier-Stokes equations
  16. ...and 6 more sections

Key Result

Lemma 1

The weighted sum $\phi(\mathrm{n}_{r}^{p},\mathrm{n}_{k}^{n})$ verifies and verifies also the following recurrence formulas:

Figures (14)

  • Figure 1: Graph visualization of the recurrence formula in \ref{['recursive_fN']} for $n=4$.
  • Figure 2: Graph representation of recurrence formula $\mathrm{u}_3$.
  • Figure 3: Figure of the graph generating the coefficients $a_{k}^{n}$ .
  • Figure 4: Comparison of the error between SPGD, STSE with $\alpha_0 = h^m$ stabilization, and STSE with $\lambda_n = \frac{2\nu\tau}{2n+1}$ stabilization. Left column: error versus mesh size $h$. Right column: error versus time step $\tau$. Top row: diffusion coefficient $\nu = 0.01$. Bottom row: diffusion coefficient $\nu = 1$. All simulations use first-order finite elements ($s=1$).
  • Figure 5: CPU time versus error for SPGD, STSE with $h^m$ stabilization, and STSE with $\lambda_n = \frac{2\nu\tau}{2n+1}$ stabilization. Left column: first-order finite elements ($s=1$); right column: second-order finite elements ($s=2$). Top row: diffusion coefficient $\nu = 0.01$; bottom row: $\nu = 1$. Simulations are performed with fixed mesh size $h = 0.01$ while varying the time step $\tau$.
  • ...and 9 more figures

Theorems & Definitions (3)

  • Remark 1
  • Lemma 1
  • Theorem 1