The lattice Boltzmann method (LBM) has emerged as a mesoscopic alternative to traditional Navier-Stokes solvers for modeling fluid dynamics offering advantages in computational efficiency, parallelization, and handling of complex boundaries. Despite these strengths, accurately reproducing compressible, shock-driven phenomena remains challenging. This study investigates the performance of LBM in simulating the Sod shock tube problem, a classical benchmark for compressible flow-using both single and double distribution function formulations across one- and two-dimensional lattice stencils. A MATLAB-based solver was developed to model the flow under isothermal conditions and compared to the analytical solution using the L2 norm error. The one-dimensional models achieved errors between 3.5% and 4.0%, while two-dimensional models produced slightly higher errors between 4.0 % and 4.5%, confirming that LBM can capture weakly compressible shock tube dynamics with strong qualitative agreement. However, achieving less than 1 % error and fully modeling compressible effects will require incorporating energy levels, energy transport, and an explicit treatment of the specific heat ratio (𝛾). This work provides a systematic assessment of LBM accuracy and stability, establishing a foundation for the future development of thermally coupled and fully compressible lattice Boltzmann solvers.