An explicit energy-conserving numerical method for equations of the form \(d^ 2x/dt^ 2=f(x)\)
For the numerical integration of initial-value problems modelling physical problems one may look for methods that are energy conserving because conservation laws are fundamental in physics. For the simple initial value problem \((1)\quad d^ 2x/dt^ 2=f(x);\quad x(0)=\alpha,\quad dx/dt|_{t=0}=\beta\) the energy conservation equation is obviously expressed as (2) \(1/2(dx/dt)^ 2+\phi (x)=1/2\beta^ 2+\phi (\alpha)\) where \(\phi\) is defined by \(-(d\phi /dx)=f(x)\). \textit{D. Greenspan} [J. Comput. Phys. 56, 28-41 (1984; Zbl 0561.65056)] developed two methods for the problem (1) with the conservation relation (2), but this methods are implicit forms and require iterative calculations. The present author suggests an explicit numerical method of energy conserving type and gives some numerical examples.
explicit energy-conserving numerical method
scientific article; zbMATH DE number 4092687