Understanding the behavior of viscous incompressible fluids is essential for scientificapplications, yet when modeling them presents significant theoretical and practical chal-lenges. This study aimed to develop a numerical solver especially for the two-dimensionalNavier-Stokes equation, tailored for modeling the dynamics of a viscous incompressiblefluid, to conserve the enstrophy. The goal was to accurately simulate a physical sys-tem, and apply numerical methods such as Runge-Kutta 4, Forward Euler’s method,and pseudo-spectral methods to construct and solve the governing Partial DifferentialEquations (PDEs). These methods were evaluated for their ability to conserve the enstrophy. Not onlyenhancing our understanding of the application of the equation in real physical systems,this research also contributes to expanding the understanding of numerical methodologiesfor complicated PDEs in physical simulations. Using the aforementioned methods, together with strategically specific initial condi-tions, it is observable that the methods are sufficient for conserving the enstrophy whendealing with only the linear part of Navier-Stokes. To improve the numerical methodsconcerning the non-linear part of the Navier-Stokes, a perturbation method was imple-mented. Outcomes from this method appear promising however, implementation andmore detailed analysis are not included in this report due to time constraints. Thisrecovery strategy represents a foundation for further exploration in further research.