Figure 1 displays the numerical computation vs the relative data (High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method) as provided in the problem statement. This map is for the U_x velocity along the centerline (x = 1/2) of the container. Note that the fluid flow will be “fully developed” for some t_final > t_crit where t_crit represents the critical point in time such that the total kinetic energy plateaus. For t_final = 30 > t_crit , the numerical result maps tightly to the relative data as we can see.

Figure 2 displays the numerical computation vs the relative data for the U_y velocity along the centerline (y = 1/2 ) of the container. I noticed that the relative data was lacking data points on 1/4 < x < 3/4 , so I decided to compute and apply polynomial interpolation. Although the interpolation matches the numerical results better, this will not be considered during the error analysis as it is not a true computation regarding the initially employed algorithm. Nonetheless, the numerical result maps tightly onto the relative data as can be seen.

In the context of my Computational Fluid Dynamics simulation of a lid-driven cavity, a comprehensive analysis of the U_x velocity, U_y velocity, and pressure profileswase conducted under two distinct scenarios: γ = 0 and γ = 1. The results of these scenarios are presented in the accompanying figures, where γ = 1 demonstrates superior performance compared to γ = 0, as discerned through an examination of the number of Poisson iterations in relation to the global time step.

Under the condition of γ = 0, a fully explicit scheme is employed, characterized by a more lenient stability condition. Conversely, the γ = 1 scenario operates within a fully implicit regime, exhibiting enhanced convergence behavior even with larger time steps. As articulated by Pozrikidis, "The restrictions on the time step of the conditionally stable explicit methods are not usually prohibitive. Implicit methods allow the use of larger time steps, but the associated error may erode the accuracy and, therefore, the physical relevance of the solution. Thus, explicit methods are the standard choice in practice." Consequently, the adoption of the fully implicit treatment (γ = 1) has resulted in a notable reduction in the required PPE iterations for convergence under our specific conditions.

The figures provided illustrate that the influence of γ contributes to the smoothing of surface plot profiles, thereby facilitating advection/convection processes. This outcome aligns with expectations, as γ introduces numerical dissipation to the advection/convection terms. The introduced dissipation serves to attenuate higher-frequency oscillations, mitigating instabilities and promoting overall smoothness in the function. Notably, this smoothing effect is most pronounced in the contour plots, where rigid profiles dissipate into coherent contour lines, as anticipated.

In conducting the convergence and error analysis, it is imperative to note that all data and plots presented are instances where t_final surpasses a critical time threshold, denoted as t_crit. Through an empirical exploration involving varied t_final values, it was established that a duration of 30 seconds proves to be a suitable choice, affording the simulation ample time for complete development.

Figure 3 encapsulates a depiction of the Poisson Iterations with respect to time, showcasing a discernible trend. Notably, the number of Poisson Iterations attains a plateau, stabilizing at approximately 350 iterations after the initial 10 seconds of simulation time. This observed behavior underscores the convergence characteristics and attests to the efficiency of the simulation process, affirming its reliability in achieving the desired accuracy.

In the course of my comparative analysis between a fully explicit and implicit method, a noteworthy revelation emerged in the form of a contour map depicting the residual. The ensuing observations are as follows in Gallery 4 where γ =0 is on the left and γ =1 on the right.

These illustrative figures offer valuable insights into the convergence behavior under both conditions. The residuals, serving as indicators of the disparity between computed values at each iteration of the Pressure-Poisson Equation (PPE) and the values that would precisely satisfy the equations within the specified tolerance. Notably, the residual for γ = 1 showcases a lower error near the middle, distinguished by the more pronounced intensity of the darker blue curve.

To be meticulous, a grid refinement was implemented, spanning from dx = 0.05 to dx = 0.025. It becomes evident that the explicit scheme experienced a degradation by a factor of 5.14 in the infinity norm, transitioning from 9.13 × 10^(−4) to 4.7 × 10^(−3). Simultaneously, the implicit scheme (γ = 0) exhibited a decline by a factor of 6.11 in the infinity norm, decreasing from 5.89 × 10^(−4) to 3.6 × 10^(−3). These findings underscore the sensitivity of both schemes to grid refinement and provide valuable insights into the trade-offs between accuracy and computational cost.