Fourth-order compact difference schemes for the one-dimensional Euler-Bernoulli beam equation with damping term

TL;DR

This paper develops a high-accuracy finite difference method using compact schemes for the damped Euler-Bernoulli beam equation, achieving fourth-order spatial accuracy and second-order temporal accuracy with proven stability and convergence.

Contribution

It introduces a novel compact finite difference scheme combined with variable substitution for efficient damping term handling, with rigorous analysis and verification.

Findings

Achieves fourth-order spatial accuracy

Proves stability and convergence theoretically

Demonstrates high accuracy and efficiency in numerical tests

Abstract

This paper proposes and analyzes a finite difference method based on compact schemes for the Euler-Bernoulli beam equation with damping terms. The method achieves fourth-order accuracy in space and second-order accuracy in time, while requiring only three spatial grid points within a single compact stencil. Spatial discretization is carried out using a compact finite difference scheme, with a variable substitution technique employed to reduce the order of the equation and effectively handle the damping terms. For the temporal discretization, the Crank-Nicolson scheme is applied. The consistency, stability, and convergence of the proposed method are rigorously proved. Numerical experiments are presented to verify the theoretical results and demonstrate the accuracy and efficiency of the method.