To correctly solve hyperbolic conservation laws, which often have shocks and discontinuities, you need high-order finite difference methods. Numerical spread and instability are problems with traditional low-order methods. This is why more advanced methods like ENO and WENO were created. These methods get very accurate results in areas with flat surfaces while stopping variations near steep slopes. This makes sure that the models are strong and work well. The study talks about the basics of math, rules for stability and consistency, the structure of algorithms, and how to treat boundaries. Numerical tests show that accuracy, shock absorption, and stability gaps are all getting better.