Numerical Integration Error in Linear Ordinary Differential Equations—Characteristic Root Distortion by Integration

Obtaining solutions to large-scale linear ordinary differential equations (ODEs) inevitably relies on numerical integration methods that have developed in various contexts to ultimately approximate the differentiation of continuous-time values using differences in discrete values. This approximation typically induces integration error, which we hypothesize as arising from the deviation of the characteristic roots from those of the original ODEs. Herein, this hypothesis was proved by numerical experiments on integration of undamped vibration equation, which yielded error amplification over long-term integration. We numerically integrated second-order single and eighth order multiple undamped vibrational equations using various formulas and analyzed the results both qualitatively and quantitatively. In the quantitative analysis, we estimated the transfer function from the integrated numerical solutions and compared its roots with those of the original equation. This approach revealed that the numerical solutions corresponded to the solution of the ODEs with roots different from those of the original ODE. For example, integrating a vibration equation with (roots) $\times $ (step width) $= 0~\pm ~1$ i using the backward Euler method resulted in solutions of the ODE with (roots) $\times $ (step width) = – $0.3466~\pm ~0.7854$ i, which were the roots of damped vibration with a lower angular frequency than that of the original ODE. This phenomenon was clearly observed for an eighth-order vibrational ODE. As the product of the roots and step width decreases, the values approach the roots of the original ODE.