Dawn of geometric integration скачать в хорошем качестве

Dawn of geometric integration

Comparative simulations using acclaimed low order explicit integrators demonstrating the errors or structure preservation associated with symplectic and non-symplectic methods. The comparisons include chronologically the first order methods: "forward Euler" and "symplectic Euler", the second order methods: "Heun's method" and "leapfrog", and the third order methods: "RK3 (Kutta's method)" and "SI3 Ruth's (1983)". 0:00 forward Euler vs symplectic Euler (order 1) 2:00 Heun's method vs leapfrog (order 2) 4:00 Kutta's method vs SI3 Ruth's (order 3) where symplectic Euler, leapfrog and SI3 Ruth's are symplectic integrators. Each pair of integration methods are applied to the Kepler problem, the simple pendulum and the Hénon-Heiles system. The symplectic methods (depicted with blue) are implemented using Hamilton's equations whereas the non-symplectic methods (depicted with red) are applied directly to the pairs of first order ODEs obtained from the second order ODEs of each system. Various reasonable time steps are examined, although only to the extend of establishing a qualitative and demonstrative presentation of the methods and their associated errors or structure preservation. Listed errors are the absolute deviation from the initial energy. Segments of published works and dynamical systems associated with the scientists are depicted as historical and chronological context. 🎵 "Trauma (Worakls Remix)" by "Worakls", original by "Anthony Favier" | not affiliated with/endorsed by.