Stabilizing the explicit euler integration of stiff and undamped linear systems

Euler's integration methods are frequently used for numerical integration as well as for real-time implementation of linear systems. However, when the integrated system is either undamped or stiff, Euler's explicit integration becomes unstable, regardless of the forcing input In this work, it is shown that this instability can be avoided by a judicious selection of the state variables. Instead of the generalized coordinates and velocities, it is proposed to define state variables based on the method of variation of parameters. It is proven that the variation-of-parameters-based state variables yield a bounded numerical error for undamped and stiff systems, provided that the forcing inputs are bounded. The analysis is performed for both deterministic and stochastic inputs. In the stochastic case, the numerically calculated covariance matrix entries diverge when using the generalized coordinates and velocities, but remain bounded when implementing the variation-of-parameters-based approach. The newly developed formalism is illustrated by a number of examples of practical interest, showing that the variation-of-parameters-based approach is also more computationally efficient than the standard approach.