A Variational Approach to Digital Integration

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Abstract
Variational equations are derived as a preliminary step in determining efficient digital integration techniques for nonlinear dynamical systems. The variational approach is applied initially to linear time-invariant systems to introduce the basic concept and then to nonlinear time-varying systems. For systems containing both linear and nonlinear parts, a combination technique which uses the exact difference equation for the linear part is developed. Higher order variational equations are also derived and compared on a simple system. Numerical approximations for solving these variational equations are discussed and illustrated for a second-order mildly nonlinear example. A significant improvement in both accuracy and execution time is realized over results obtained by the conventional fourth-order Runge–Kutta method. Finally, the new approach is discussed from the viewpoint of computational experience and special limitations for practical applications.