Abstract: The problem of preserving fidelity both geometrically and dynamically in scientific numerical solutions of nonlinear ordinary differential and partial differential evolution equations is studied from the point of view of preserving local differential structure and approximating global integration structure of the dynamical system. The ordinary differential equations are lifted to the partial differential equations and the partial differential equations are lifted to the functional differential equations in the framework of algebraic dynamics. The local differential structure and global integration structure of the dynamical system are described in terms of Lie algebra and Lie group. The time evolution of the differential dynamic system is described locally by the time translation operator ( Lie algebra ) and globally by the time evolution operator ( Lie group ). The exact analytical piece-like solution of the dynamical equation is obtained in term of Talyor series with a local convergent radius. A new algorithm―algebraic dynamics algorithm with a controllable precision better than Runge Kutta Algorithm, Symplectic Geometric Algorithm and others, is proposed based on the finite order truncation of the exact analytical solutions. The advantage and accuracy of the new algorithm are compared with Runge Kutta Algorithm, Symplectic Geometric Algorithm, and other available algorithms in a series of typical computer numerical experiments. |