The application of finite-difference methods to initial-value problems, with emphasis on parabolic equations, is considered using the one- and two-dimensional unsteady diffusion equations as model problems.Single-step methods are introducted for ordinary differential equations, and more general explicit and implicit methods are articulated for partial differential equations.Numerical stability analysis is covered using the matrix method, von Neumann method, and the modified wavenumber method.These numerical methods are also applied to nonlinear convection problems.A brief introduction to numerical methods for hyperbolic equations is provided, and parallel computing is discussed.