Observations of the Sun provide unique insights into its structure, evolution, and activity, with significant implications for space weather forecasting and solar energy technologies. Ground-based telescopes offer cost-effective and flexible solutions for high-resolution solar observations, but image quality can be affected by atmospheric turbulence. Adaptive optics (AO) systems equipped with Shack-Hartmann wave front sensors (SH-WFS) enable real-time image correction to mitigate these effects. The accuracy of SH-WFS relies on correlation algorithms that measure wave front shifts, but reaching consistent conclusions regarding their accuracy remains challenging. In this study, we conducted an evaluation and comparison of standard correlation algorithms (the Square Difference Function, Normalized Cross-Correlation, Absolute Difference Function, Absolute Difference Function-Squared, and the Covariance Function in the frequency domain (CFF)) using simulated and authentic solar images. We optimized the algorithms through pre-processing techniques and carefully selected the most suitable window function for the CFF algorithm. Additionally, we analyzed the influence of various factors, such as shift ranges, bias, and the size of live images on the accuracy of algorithms. The consistent findings revealed that the CFF algorithm demonstrates superior measurement accuracy and robustness compared to the others. Choosing the CFF algorithm for solar observations can significantly enhance measurement accuracy, AO system performance, and the overall quality of solar research findings, thereby providing crucial support for space weather forecasting and other related scientific fields.