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SUMMARYNewton's method is developed for solving the 2-D Euler equations. The Euler equations are discretized using a ÿnite-volume method with upwind ux splitting schemes. Both analytical and numerical methods are used for Jacobian calculations. Although the numerical method has the advantage of keeping the Jacobian consistent with the numerical residual vector and avoiding extremely complex analytical di erentiations, it may have accuracy problems and need longer execution time. In order to improve the accuracy of numerical Jacobians, detailed error analyses are performed. Results show that the ÿnite-di erence perturbation magnitude and computer precision are the most important parameters that a ect the accuracy of numerical Jacobians. A method is developed for calculating an optimal perturbation magnitude that can minimize the error in numerical Jacobians. The accuracy of the numerical Jacobians is improved signiÿcantly by using the optimal perturbation magnitude. The e ects of the accuracy of numerical Jacobians on the convergence of the ow solver are also investigated. In order to reduce the execution time for numerical Jacobian evaluation, ux vectors with perturbed ow variables are calculated only for neighbouring cells. A sparse matrix solver that is based on LU factorization is used. E ects of di erent ux splitting methods and higher-order discretizations on the performance of the solver are analysed.