Finite difference method for nonlinear damped viscoelastic Euler-Bernoulli beam model

TL;DR

This paper develops and analyzes a finite difference numerical scheme for a nonlinear damped viscoelastic Euler-Bernoulli beam model, establishing stability, error estimates, and solution existence.

Contribution

It introduces a combined spatial-temporal discretization approach and provides rigorous stability, error analysis, and proof of solution existence for the nonlinear model.

Findings

Numerical solutions are stable over long times.

Error estimates are derived for the fully-discrete scheme.

Numerical results confirm theoretical predictions.

Abstract

We propose and analyze the numerical approximation for a viscoelastic Euler-Bernoulli beam model containing a nonlinear strong damping coefficient. The finite difference method is used for spatial discretization, while the backward Euler method and the averaged PI rule are applied for temporal discretization. The long-time stability and the finite-time error estimate of the numerical solutions are derived for both the semi-discrete-in-space scheme and the fully-discrete scheme. Furthermore, the Leray-Schauder theorem is used to derive the existence and uniqueness of the fully-discrete numerical solutions. Finally, the numerical results verify the theoretical analysis.