Many physical systems described by an initial-value problem for a system of ordinary differential equations (ODEs) conserve physical quantities, such as the net charge or total energy, as the system evolves. Typical codes for the numerical solution of the ODEs will not conserve these quantities, and this can lead to solutions which are not even qualitatively correct. One way to impose conservation laws is to perturb the numerical solution at each step of the integration. A simple theory is developed in this paper which tells how this should be done so as to guarantee convergence of codes based on one-step methods. It is also easy to interpret the effect on the accuracy of the perturbations. © 1986.
Shampine, L. F. (1986). Conservation laws and the numerical solution of ODEs. Computers and Mathematics with Applications, 12(5-6 PART 2), 1287–1296. https://doi.org/10.1016/0898-1221(86)90253-1