The problem of preserving fidelity in numerical computation of nonlinear ordinary differential equations is studied in terms of preserving local differential structure and approximating global integration structure of the dynamical system. The ordinary differential equations are lifted to the corresponding partial differential equations in the framework of algebraic dynamics, and a new algorithm—algebraic dynamics algorithm is proposed based on the exact analytical solutions of the ordinary differential equations by the algebraic dynamics method. In the new algorithm, the time evolution of the ordinary differential system is described locally by the time translation operator and globally by the time evolution operator. The exact analytical piece-like solution of the ordinary differential equations is expressed in terms of Taylor series with a local convergent radius, and its finite order truncation leads to the new numerical algorithm with a controllable precision better than Runge Kutta Algorithm and Symplectic Geometric Algorithm.