The goal of this study is to extend the modified equation technique to the Burgers’ equation.The modified equation is a technique for evaluating various qualities or properties of a finite-difference analogue of a given partial differential equation. These qualities include order of accuracy, consistency, stability, dissipation, and dispersion. Our approach is based on the semi-Lagrangian formulation. The essential idea of the semi-Lagrangian formulation is to combine the Eulerian derivative and the convective term into a Lagrangian derivative. Another goal of this study is to attempt to develop highly accurate and efficient semi-Lagrangian finite difference methods. A new designing algorithm with the aid of the modified equation theory and a new solutions to the considerid equations have been introduced. The accuracy of the proposed semi-Lagrangian finite difference methods are surveyed through careful error analysis. We show that the overall accuracy of the proposed semi-Lagrangian schemes depends on two factors: one is the global truncation error which can be obtained by the modified equation analysis and the other is a generic feature of semi-Lagrangian methods which characterize their non-monotonic dependence on the time step-size. In this thesis, we inrestigate the semi-Lagrangian methods and modified equation technique for solving evolutionary partial differential equations. First of all we provide a collection of the required prerequisites including the process of obtaining modified equation, line interpolation and also a brief description of the characteristics and the locan one-dimensional (LOD) method, to better understand the thesis implications. We also verify the high accuracy and unconditional stability of the five-point implicit scheme by conducting numerical simulation to the nonlinear Burgers’ equation which is a canonical nonlinear model to test numerical methods. Then the semi-Lagrangian formulation of this equation is discretized along the characteristic curve and we have used the second-order Runge–Kutta method to locate the departure points. The solution values at the departure points are obtained by interpolation. Therefore, we have the interpolative error to obtain the global truncation error of the difference scheme using the modified equation technique. An error analysis is conducted for the proposed difference schemes, followed by numerical experiments for verification of the analytical results. The results of this study suggests a direction where more accurate and efficient semi-Lagrangian methods can be developed, and where the be easily extended to nonlinear applications. In the following we have implemented semi-Lagrangian and modified equation approaches on 2D-Burgers’ equation and define a six-point explicit scheme, a six-point implicit scheme and a ten-point implicit scheme for these equations with the aid of a LOD method to split the implicit schemes. Next we have focused on the numerical solution of the KdV and mKdV equations using the semi-Lagrangian and modified equation approaches. Followed by numerical tests for verification of the analytical results. Finaly in the appendix, we have described details to obtain the modified equation carried out with the Mathematica package.