Unified Framework for Evaluating Numerical Integration Methods through Characteristic Root Distortion in Linear Ordinary Differential Equations

Analytically solving complex or large-scale differential equations is often difficult or even impossible, making numerical integration methods indispensable. However, as all numerical integration methods approximate a continuous-time equation using a discrete-time model, they inevitably introduce errors into the numerical solution. This study presents a unified framework for evaluating the numerical errors introduced by various integration methods. The integration formulae are expressed as equations involving differential and difference operators applied to the solution of the equation. This study applies integration methods to constant-coefficient linear ordinary differential equations (ODEs) and adopt an approach in which these expressions are transformed using Laplace and z-transforms, deriving characteristic equations that capture the essential properties of widely used integration methods. Based on the characteristic equations, distortions of the characteristic root —real and conjugate complex values that represent the fundamental properties of linear ODEs due to integration methods have been evaluated, and regions where stable roots become numerically unstable have also been identified. Additionally, transfer functions were estimated from the numerical solutions, and the distorted characteristic roots were corrected using the inverse of the approximation formula. The proximity of the restored roots to the original roots served as a quantitative indicator of the accuracy and reliability of each integration method. This unified evaluation approach revealed the distinct characteristics of widely used integration methods and offered a generalizable basis for interpreting errors in numerical solutions.