The mathematical model describing transients in natural gas pipelines constitutes a non-homogeneous system of non-linear hyperbolic conservation laws. The time splitting approach is adopted to solve this non-homogeneous hyperbolic model. At each time step, the non-homogeneous hyperbolic model is split into a homogeneous hyperbolic model and an ODE operator. An explicit 5-point, second-order-accurate total variation diminishing (TVD) scheme is formulated to solve the homogeneous system of non-linear hyperbolic conservation laws. Special attention is given to the treatment of boundary conditions at the inlet and the outlet of the pipeline. Hybrid methods involving the Godunov scheme (TVD/Godunov scheme) or the Roe scheme (TVD/Roe scheme) or the Lax -Wendroff scheme (TVD/LW scheme) are used to achieve appropriate boundary handling strategy. A severe condition involving instantaneous closure of a downstream valve is used to test the efficacy of the new schemes. The results produced by the TVD/Roe and TVD/Godunov schemes are excellent and comparable with each other, while the TVD/LW scheme performs reasonably well. The TVD/Roe scheme is applied to simulate the transport of a fast transient in a short pipe and the propagation of a slow transient in a long transmission pipeline. For the first example, the scheme produces excellent results, which capture and maintain the integrity of the wave fronts even after a long time. For the second example, comparisons of computational results are made using different discretizing parameters.