Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Numerical Treatment of an Abstract Nonlinear System of Coupled Hyperbolic Equations Associated with the Timoshenko Model

Jemal Rogava, Zurab Vashakidze

TL;DR

This work develops and analyzes a symmetric three-layer semi-discrete time-stepping scheme for an abstract nonlinear, coupled hyperbolic system modeling Timoshenko-type beams. By evaluating the nonlinear term at the temporal midpoint, each time step reduces to a linear problem that can be solved in parallel, and the authors prove local-in-time second-order convergence. They then extend the framework to a spatially one-dimensional nonlinear dynamic Timoshenko model and employ a Legendre–Galerkin spectral method in space, yielding sparse, decoupled tridiagonal systems that facilitate efficient computation. Comprehensive a priori estimates, error analyses, and four numerical benchmarks validate the method’s stability, accuracy, and practicality, with open-source code provided for replication and exploration.

Abstract

The present work addresses the Cauchy problem for an abstract nonlinear system of coupled hyperbolic equations associated with the Timoshenko model in a real Hilbert space. Our purpose is to develop and delve into a temporal discretization scheme for approximating a solution to this problem. To this end, we propose a symmetric three-layer semi-discrete time-stepping scheme in which the nonlinear term is evaluated at the temporal midpoint. As a result, at each time step, this approach reduces the original nonlinear problem to a linear one and enables parallel computation of its solution. Convergence is proved, and second-order accuracy with respect to the time-step size is established on a local temporal interval. The proposed scheme is then applied to a spatially one-dimensional nonlinear dynamic Timoshenko beam system, and the results obtained for the abstract nonlinear system are extended to this setting. A Legendre-Galerkin spectral approximation is employed for the spatial discretization. By taking differences of Legendre polynomials within the Galerkin framework, the resulting linear system is sparse and can be efficiently decoupled. The convergence of the method is also investigated. Finally, several numerical experiments on carefully chosen benchmark problems are conducted to validate the proposed approach and to confirm the theoretical findings.

On the Numerical Treatment of an Abstract Nonlinear System of Coupled Hyperbolic Equations Associated with the Timoshenko Model

TL;DR

This work develops and analyzes a symmetric three-layer semi-discrete time-stepping scheme for an abstract nonlinear, coupled hyperbolic system modeling Timoshenko-type beams. By evaluating the nonlinear term at the temporal midpoint, each time step reduces to a linear problem that can be solved in parallel, and the authors prove local-in-time second-order convergence. They then extend the framework to a spatially one-dimensional nonlinear dynamic Timoshenko model and employ a Legendre–Galerkin spectral method in space, yielding sparse, decoupled tridiagonal systems that facilitate efficient computation. Comprehensive a priori estimates, error analyses, and four numerical benchmarks validate the method’s stability, accuracy, and practicality, with open-source code provided for replication and exploration.

Abstract

The present work addresses the Cauchy problem for an abstract nonlinear system of coupled hyperbolic equations associated with the Timoshenko model in a real Hilbert space. Our purpose is to develop and delve into a temporal discretization scheme for approximating a solution to this problem. To this end, we propose a symmetric three-layer semi-discrete time-stepping scheme in which the nonlinear term is evaluated at the temporal midpoint. As a result, at each time step, this approach reduces the original nonlinear problem to a linear one and enables parallel computation of its solution. Convergence is proved, and second-order accuracy with respect to the time-step size is established on a local temporal interval. The proposed scheme is then applied to a spatially one-dimensional nonlinear dynamic Timoshenko beam system, and the results obtained for the abstract nonlinear system are extended to this setting. A Legendre-Galerkin spectral approximation is employed for the spatial discretization. By taking differences of Legendre polynomials within the Galerkin framework, the resulting linear system is sparse and can be efficiently decoupled. The convergence of the method is also investigated. Finally, several numerical experiments on carefully chosen benchmark problems are conducted to validate the proposed approach and to confirm the theoretical findings.
Paper Structure (15 sections, 13 theorems, 151 equations, 9 figures)

This paper contains 15 sections, 13 theorems, 151 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Formulation of the Problem
  3. Outline of the Problem
  4. Symmetric Three-Layer Semi-Discrete Scheme
  5. Preliminaries
  6. Uniform Boundedness of the Solution to a Discrete Problem and Its Corresponding Difference Analogues of the Derivative
  7. Uniform Boundedness of High-Order Terms Corresponding to the Solution of a Discrete Problem
  8. Error Estimates for Approximate Solutions of the System
  9. Spatial One-Dimensional Nonlinear Dynamic Timoshenko Model
  10. Semi-Discrete Scheme
  11. Reduction of a Semi-Discrete Scheme to an Intermediate System
  12. Legendre–Galerkin Spectral Method
  13. Some Notes on Solving the Derived Subsystems of the Galerkin Linear Equations
  14. Estimate of the Error in the Legendre–Galerkin Spectral Method
  15. Illustrative Numerical Results

Key Result

Lemma 3.1

Consider the sequences of nonnegative numbers ${\left\{ {\alpha}_{k} \right\}}_{k = 0}^{n}$ and ${\left\{ {c}_{k} \right\}}_{k = 0}^{n}$, which satisfy the following inequality: where ${s} > 0$ and ${\tau} > 0$ hold. Consequently, the estimate is valid

Figures (9)

  • Figure 1: Comparison between the exact solutions and their numerical approximations at the final time layer, where the number of trial functions is specified as $N = 20$.
  • Figure 2: Comparison of the exact solutions and their numerical approximations at the final temporal layer using $N = 35$ approximation basis functions, along with the evolution of the associated errors across all time layers.
  • Figure 3: Exact solutions and their numerical approximations at the final time instant $t = 1$, computed with $N = 29$ trial functions.
  • Figure 4: Exact and numerical solutions at the final temporal layer for $N = 45$ trial functions, and evolution of the errors $E_{1,k}$ and $E_{2,k}$ across all time layers, illustrating the accuracy of the combined numerical scheme.
  • Figure 5: Exact solutions together with their corresponding numerical approximations at the final time instant $t = 4$, obtained using $N = 7$ trial functions.
  • ...and 4 more figures

Theorems & Definitions (30)

  • Lemma 3.1: For further details, refer to Lemma 3.2 in RogavaTsiklauri2012
  • Lemma 3.2: See Lemma 3.1 in RogavaTsiklauri2014 for further details
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • Remark 3.7
  • Remark 3.8
  • Lemma 4.1
  • Lemma 5.1
  • ...and 20 more