Conserving Numerical Methods for x = f(x)

Abstract

**Please note that the full text is embargoed** ABSTRACT: Physics is characterized by conservation laws and by symmetry [1]. Unfortunately, the application of numerical methodology in approximating solutions of initial value problems usually does not preserve either of these invariants. In this sense, the use of a computer destroys the physics of a dynamical model. We will show here how to conserve total energy when solving the nonlinear initial value problem [see pdf for notation] on a computer. Moreover, the energy conserved will be exactly that of (1.1), not a new "energy" which is defined by the numerical method (see, e.g., Langdon [5]). Two distinctly different methods will be developed, one of which is completely conservative and symmetric, the other of which reveals how to convert any numerical method to an energy conserving one.