We have carried out high precision orbit calculation, by using various numerical techniques with accuracy of higher than fourth order, in order for exact prediction on position and velocity of celestial bodies and artificial satellites. General second order ordinary differential equation has been solved numerically to test the performance for each of numerical methods. We have compared computed values with exact solution obtained by using universal variables for two body problem and discussed overall results of numerical methods used in our calculation. As a result, it is found that high order difference table method called as Gauss-Jackson method is best one with easiness and efficiency in the increase of accuracy by number of initial values.