This thesis delves deeply into numerical solutions to nonlinear conservation laws. It focuses largely on the Method of Characteristics, its application in solving conservation laws, and the implementation of solutions in MATLAB using the Lax-Friedrichs scheme. The first chapter offers numerous examples of linear conservation rules and examines its numerical scheme. The equations’ stability qualities are care- fully examined. In this chapter, the Method of Characteristics is extensively used to solve conservation laws, and the efficiency of the Upwind Scheme is proved. The generic solution to the nonlinear conservation law, u_t + f (u)_x = 0, is investigated in Chapter 2. The Characteristics Method is used to deduce the criteria for u_t + f (u)_x = 0 and to examine the Lax-Friedrichs scheme. The Rankine-Hugoniot condition, the development of similarity and shock wave solutions, and the distinctions between convex and concave flux are all cov- ered in this chapter. It also provides a thorough comparison of the Method of Characteristics and the Finite Difference Technique. The chapter concludes with an examination of inadequate conservation legislation remedies. The third chapter tackles a more complicated Riemann issue, presenting a thorough solution as well as MATLAB-based visualization. The use of prior chapters’ knowledge and methodologies to this more complicated issue illus- trates the methods’ adaptability and robustness. In conclusion, this thesis makes an important addition to the understand- ing and application of the Method of Characteristics and the Lax-Friedrichs scheme in the solution of nonlinear conservation laws, as evidenced by prac- tical MATLAB implementation. The comparison of numerical systems, as well as the expansion to complicated Riemann problems, improve the study’s value in improving numerical analysis in general