This thesis describes numerical methods for the solution of advection equations in one-dimensional space. The methods are based on combining the multigrid and cubic spline collocation (CSC) methodologies. Multigrid methods for cubic splines are presented. Appropriate restriction and extension operators are developed for cubic splines that satisfy various boundary conditions. The multigrid methods are applied to CSC equations arising from the discretization of one-dimensional second-order differential equations. The rate of convergence are proved. Multigrid methods for cubic splines are then extended to the solution of one-dimensional shallow water equations (SWEs). The SWEs are discretized in time with a semi-Lagrangian semi-implicit scheme and in space with CSC methods. We discuss three different discretization approaches at each time step and develop new numerical methods. Through comparison with Jacobi's iterative method and convergence discussion, we show that our multigrid methods for CSC are convergent and efficient. The numerical results confirm our analysis.