This book deals with the derivation of diagonally implicit Runge-Kutta (DIRK) methods of order four and five which are specially designed for the integration of linear ordinary differential equations (LODEs). The restriction to LODEs with constant coefficients reduces the number of order equations which the coefficients of Runge-Kutta (RK) methods must satisfy. The coefficients of the RK methods are chosen such that the error norm is minimized, this resulted in methods which are almost one order higher than the actual order. The stability polynomials and stability regions of the methods are then obtained using MATHEMATICA package. Codes using C++ programming based on the methods are developed to test sets of problems on linear ordinary differential equations. Numerical results show that the new methods are more efficient than the existing methods.