The last few decades have seen significant efforts to model erosion, transport and deposition of sediment in flowing surface water. A wide variety of models, ranging from simple empirical models to highly sophisticated physically-based models have been presented. This thesis aims to present a physically realistic but simple model to simulate sediment routing. The model uses shallow water equations for hydrodynamics, while it is coupled with sediment conservation and transport laws to enable modification of the underlying substrate. To solve the model, I present a straightforward Riemann-solver free approach, based on the explicit non-oscillatory central differencing (NOC) scheme. The used central approach becomes excessively diffusive when applied to highly nonlinear advection problems. In this thesis, I present a correction to reduce such numerical dissipation. To test the accuracy and robustness of the scheme, I present a number of test cases and the results are compared with analytical solutions and laboratory experiments.