For one-dimensional parabolic equations, the forward Euler scheme is simple to implement; however, its step ratio is restricted. The backward Euler scheme and the Crank-Nicolson scheme are both unconditionally stable, and the computational cost is not large when the double sweep method is applied. For high-dimensional parabolic equations, the forward Euler scheme is easy to implement, but its stability condition is more stringent than that in the one-dimensional case. Although the backward Euler and Crank-Nicolson schemes are unconditionally stable, the size of the system of difference equations at each time level becomes larger, and the system is no longer a tridiagonal system of linear equations. Solving such a system requires considerable CPU time. Similar challenges can be found for hyperbolic equations. Therefore, it is necessary to explore new unconditionally stable difference schemes with reduced computational complexity. The alternative directional implicit (ADI) scheme introduced in this chapter is unconditionally stable and can be solved efficiently using the double sweep method.