In this article, numerical solutions of the seven different forms of the single soliton and double soliton solutions of the Korteweg-de Vries equation are investigated. Since numerical solution of the six test problems for small-times do not exist in the literature, the present numerical results firstly are reported with exact solutions. Besides small-time solutions, long-time solutions of all test problems are obtained and compared with some of the earlier works. Present algorithm which is based on combination of finite difference method and differential quadrature method have obtained superior results than those in the given literature. Numerical and exact solutions for small-time of all test problems are plotted together for all test problems. Since the numerical results are too close to exact solutions the graphs are indistinguishable. Numerical simulations for long-time solutions are plotted and the error graphs are plotted for the end of the simulations of all test problems.