## 5 Buoyancy-Driven Square Cavity
### Residual Convergence Analysis  
The baseline test case is run in a \(70\times 70\) uniform mesh with \(\Delta t=0.1\), a Generalized Minimal Residual (GMRES) tolerance of \(10^{-8}\) with a maximum of 60 iterations with 9 restarts and an ILU preconditioner. The classic tau allowed a larger time-step for \(\epsilon=0.6\)  
Figure 1: Buoyancy-driven square cavity. Problem setup.  
so a value of \(\Delta t=1.0\) was chosen for this case. The viscosity has been assumed to be constant.  
Figure 2 compares the residual convergence for various stabilization matrices for the full Navier-Stokes equations. One can conclude that the diagonal tau should be avoided due to slow convergence or even divergence for large \(\epsilon\). This result stresses the importance of off-diagonal terms in the stabilization matrix, which is in agreement with [13; 35]. For any \(\epsilon\), the HH, Polner, and compressible Polner taus show a similar behavior. Regarding the classic stabilization matrix, it is observed that the residual converges more slowly than the method with the incompressible taus.  
Figure 3 compares, for the Boussinesq model and the SUPG finite element method, the residual convergence of the incompressible stabilization matrices. From the results, one can conclude that the diagonal tau should be avoided, since, for the given computing parameters, it does not converge for small \(\epsilon\). Again, this result stresses the importance of off-diagonal terms [13; 35]. For any \(\epsilon\), the HH and Polner taus show a similar behavior. For the incompressible model, recall that the compressible Polner tau is the same as the Polner tau.  
The Boussinesq model solved with the SGS finite element method is evaluated in Figure 4, which compares the residual convergence for the incompressible stabilization matrices. Again, the diagonal tau should be avoided, since, for the given computing parameters, it shows no converge for small \(\epsilon\), stressing the importance of off-diagonal terms [13; 35]. For small \(\epsilon\), the Polner tau presents a better convergence, whereas for large \(\epsilon\), both Polner and HH taus behave similarly. As before, for the incompressible model, the Polner tau is the same as the compressible Polner tau.  
Figure 3: Buoyancy-driven square cavity. Boussinesq model with SUPG. Residual convergence for various stabilization matrices  
Figure 2: Buoyancy-driven square cavity. Full Navier-Stokes equations. Residual convergence for various stabilization matrices  
Figure 5 compares, for the Boussinesq model and both the SUPG and SGS methods, the effect of GMRES tolerance. The Boussinesq model allows further CPU time savings by increasing the GMRES tolerance. This is not the case for the full Navier-Stokes model, which requires a tolerance of \(10^{-8}\).  
Finally, Figure 6 compares the residual convergence for various models and methods. For the Boussinesq model, both SUPG and SGS with either the HH and the Polner taus, behaved similarly; SUPG perhaps showed faster convergence than SGS for small \(\epsilon\). Regarding the full Navier-Stokes equations, both HH and the Polner tau behave identically and produce a faster convergence than the classic tau, although the classic tau improves convergence as the temperature difference is increased. Note that, for the same computing parameters and small \(\epsilon\), the Boussinesq model converges faster than the full Navier-Stokes equations. This advantage tends to disappear as \(\epsilon\) is increased.  
Figure 4: Buoyancy-driven square cavity. Boussinesq model with SGS. Residual convergence for various stabilization matrices  
Figure 5: Buoyancy-driven square cavity. Residual convergence for GMRES tolerances for the Boussinesq model.