The NS PDEs have been modified to enhance numerical tractability, followed by corrective iterations to recover solutions for the NS PDEs. It is shown that the scheme can be transformed into the implicit upwind approximation, yielding an identical solution. The iterative scheme guarantees convergence, a predictable number of iterations, and provides initial rapid convergence. The number of iterations depends on the size of mesh. In triangular matrix regions, iterations can be accelerated using a sweeping approach similar to the fast-marching method. Outside triangular submatrices, the iterative process is retained. This method is proposed to facilitate cross-validation with a reference solution.