A Galerkin/POD ReducedOrder Model from Eigenfunctions of NonConverged Time Evolution Solutions in a Convection Problem
Abstract
:1. Introduction
2. Formulation of the Problem
Stationary Solutions and Linear Stability Analysis
3. The POD ReducedOrder Method
Algorithm 1: Calculation of the snapshots. 

Algorithm 2: Computation of ${\lambda}_{i}$ and POD modes. 

 In practice, matrices ${\mathcal{S}}^{P}$ and ${\mathcal{M}}^{P}$ are not considered and Algorithm 2 is not applied to the pressure field because the pressure field disappears from the variational formulation used to solve the problem; see the next section.
 The POD bases for velocity fields ${B}_{u,I}^{POD}$ and ${B}_{w,I}^{POD}$ are computed, separately, following Algorithm 2. Below, we refer to the POD basis for the velocity field $\mathit{v}$ as ${B}_{\mathit{v},I}^{POD}$. Each element of ${B}_{\mathit{v},I}^{POD}$, ${\mathcal{Q}}_{\mathit{v}}^{j}$, is obtained concatenating vertically the corresponding elements in ${B}_{u,I}^{POD}$ and ${B}_{w,I}^{POD}$, and ${\mathcal{Q}}_{u}^{j}$ and ${\mathcal{Q}}_{w}^{j}$.
3.1. The POD/Galerkin Projection Procedure for the Stationary Problem
3.2. Linear Stability Analysis of POD/Galerkin Solutions
4. Numerical Results
4.1. First Bifurcation
Validation and Results
4.2. Second Bifurcation
Validation and Results
4.3. Third Bifurcation
Validation and Results
4.4. Bifurcation Diagram
4.5. Computational Cost
 Step 5: Solve, numerically, the linear stability analysis for K transitory states:$K\xb7O\left({N}^{3}\right)$.
 Step 3: Construct the matrices ${\mathcal{M}}^{\theta},\phantom{\rule{0.166667em}{0ex}}{\mathcal{M}}^{u}$, and ${\mathcal{M}}^{w}$ from thermal and hydrodynamic snapshots: $3K{N}^{\ast}$;
 Step 4: Apply a SVD decomposition to ${\mathcal{M}}^{\theta},{\mathcal{M}}^{u}$, and ${\mathcal{M}}^{w}$ to obtain their eigenvectors and singular values: $3\xb7O\left({N}^{\ast 3}\right)$;
 Step 5: Obtain the thermal and hydrodynamic POD modes related to J and I unsaturated singular values, ${B}_{\theta ,J}^{POD\ast}=\{{\mathcal{Q}}_{\theta}^{1\ast},{\mathcal{Q}}_{\theta}^{2\ast},\dots ,{\mathcal{Q}}_{\theta}^{J\ast}\}$, ${B}_{u,I}^{POD\ast}=\{{\mathcal{Q}}_{u}^{1\ast},{\mathcal{Q}}_{u}^{2\ast},\dots ,$${\mathcal{Q}}_{u}^{I\ast}\}$, and ${B}_{w,I}^{POD\ast}=\{{\mathcal{Q}}_{w}^{1\ast},{\mathcal{Q}}_{w}^{2\ast},\dots ,{\mathcal{Q}}_{w}^{I\ast}\}$, as linear combinations of thermal and hydrodynamic snapshots: $(J+2I)K{N}^{\ast}$;
 Step 6: Orthonormalize the bases ${B}_{\theta ,J}^{POD\ast}$, ${B}_{u,I}^{POD\ast}$, and ${B}_{w,I}^{POD\ast}$ by applying a Gram–Schmidt method: $(J(J+1)+2I(I+1)){N}^{\ast}$.
 $O\left({N}_{N}{N}_{R}{N}^{\ast 2}\right)$;
 ${N}_{R}\xb7{N}_{N}\xb7O{(I+J)}^{2}$;
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
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Cortés, J.; Herrero, H.; Pla, F. A Galerkin/POD ReducedOrder Model from Eigenfunctions of NonConverged Time Evolution Solutions in a Convection Problem. Mathematics 2022, 10, 905. https://doi.org/10.3390/math10060905
Cortés J, Herrero H, Pla F. A Galerkin/POD ReducedOrder Model from Eigenfunctions of NonConverged Time Evolution Solutions in a Convection Problem. Mathematics. 2022; 10(6):905. https://doi.org/10.3390/math10060905
Chicago/Turabian StyleCortés, Jesús, Henar Herrero, and Francisco Pla. 2022. "A Galerkin/POD ReducedOrder Model from Eigenfunctions of NonConverged Time Evolution Solutions in a Convection Problem" Mathematics 10, no. 6: 905. https://doi.org/10.3390/math10060905