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On Convergence of a Three-Layer Semi-Discrete Scheme for the Nonlinear Dynamic String Equation of Kirchhoff-Type with Time-Dependent Coefficients

Jemal Rogava, Zurab Vashakidze

TL;DR

This work analyzes the Cauchy IBVP for a nonlinear Kirchhoff string with time-dependent coefficients and develops a symmetric three-layer semi-discrete scheme in time that yields linear ODEs per step by evaluating the nonlinearity at the mid-time node. A rigorously grounded error analysis, including a discrete Grönwall-type inequality and energy estimates, proves local quadratic convergence in time, while a spatial fourth-order finite-difference discretization ensures high spatial accuracy. The resulting linear, diagonally dominant, tridiagonal systems are shown to be stable, with explicit condition-number bounds that guide the choice of spatial and temporal grid parameters. Numerical experiments using GNU Octave validate the theoretical convergence rates and stability claims across multiple test problems, and the authors provide open-source code for reproducibility and SEO-friendly indexing.

Abstract

This paper considers the Cauchy problem for the nonlinear dynamic string equation of Kirchhoff-type with time-varying coefficients. The objective of this work is to develop a time domain discretization algorithm capable of approximating a solution to this initial-boundary value problem. To this end, a symmetric three-layer semi-discrete scheme is employed with respect to the temporal variable, wherein the value of a nonlinear term is evaluated at the middle node point. This approach enables the numerical solutions per temporal step to be obtained by inverting the linear operators, yielding a system of second-order linear ordinary differential equations. Local convergence of the proposed scheme is established, and it achieves quadratic convergence regarding the step size of the discretization of time on the local temporal interval. We have conducted several numerical experiments using the proposed algorithm for various test problems to validate its performance. It can be said that the obtained numerical results are in accordance with the theoretical findings.

On Convergence of a Three-Layer Semi-Discrete Scheme for the Nonlinear Dynamic String Equation of Kirchhoff-Type with Time-Dependent Coefficients

TL;DR

This work analyzes the Cauchy IBVP for a nonlinear Kirchhoff string with time-dependent coefficients and develops a symmetric three-layer semi-discrete scheme in time that yields linear ODEs per step by evaluating the nonlinearity at the mid-time node. A rigorously grounded error analysis, including a discrete Grönwall-type inequality and energy estimates, proves local quadratic convergence in time, while a spatial fourth-order finite-difference discretization ensures high spatial accuracy. The resulting linear, diagonally dominant, tridiagonal systems are shown to be stable, with explicit condition-number bounds that guide the choice of spatial and temporal grid parameters. Numerical experiments using GNU Octave validate the theoretical convergence rates and stability claims across multiple test problems, and the authors provide open-source code for reproducibility and SEO-friendly indexing.

Abstract

This paper considers the Cauchy problem for the nonlinear dynamic string equation of Kirchhoff-type with time-varying coefficients. The objective of this work is to develop a time domain discretization algorithm capable of approximating a solution to this initial-boundary value problem. To this end, a symmetric three-layer semi-discrete scheme is employed with respect to the temporal variable, wherein the value of a nonlinear term is evaluated at the middle node point. This approach enables the numerical solutions per temporal step to be obtained by inverting the linear operators, yielding a system of second-order linear ordinary differential equations. Local convergence of the proposed scheme is established, and it achieves quadratic convergence regarding the step size of the discretization of time on the local temporal interval. We have conducted several numerical experiments using the proposed algorithm for various test problems to validate its performance. It can be said that the obtained numerical results are in accordance with the theoretical findings.
Paper Structure (14 sections, 5 theorems, 68 equations, 4 figures, 5 tables)

This paper contains 14 sections, 5 theorems, 68 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Formulation of the Problem
  3. Historical Background, Physical Relevance, and Applications of the Problem
  4. Survey of Analytical and Numerical Frameworks of the Problem
  5. Outline of the Problem
  6. Description of the Time Domain Discretization Algorithm
  7. Error Estimation of the Approximate Solution Resulting from the Time Domain Discretization
  8. Auxiliary Lemmata
  9. Main Lemmata
  10. Error Estimate of the Approximate Solution
  11. Results of Numerical Computations
  12. Designing a Spatial Fourth-Order Accuracy Three-Term Difference Scheme for Solving the Obtained Linear Ordinary Differential Equations
  13. Stability of the Three-Point System Obtained by the Spatial Discretization Algorithm
  14. Numerical Illustrations

Key Result

Lemma 3.1

Let $\left\{ {\varepsilon}_{k} \right\}$, $\left\{ {a}_{k} \right\}$ and $\left\{ {h}_{k} \right\}$ be sequences of nonnegative real numbers that satisfy the following inequality then

Figures (4)

  • Figure 1: A scatter plot on a log-log scale graph representing the dependence of the maximum absolute errors between the exact and approximate solutions of \ref{['problem:test1']} on the temporal steps, along with a regression line.
  • Figure 2: The exact and approximate solutions of \ref{['problem:test2']} at the point $t = 1$ are represented by solid and dashed lines, respectively.
  • Figure 3: Graph of the initial condition ${\psi}_{0}\left( x \right)$.
  • Figure 4: Graph plots illustrate the numerical solutions to the given problem at four specific temporal instances, where the error tolerance is set to $tol = {10}^{-6}$.

Theorems & Definitions (9)

  • Lemma 3.1: Discrete Grönwall-type inequality
  • Remark 3.2
  • Lemma 3.3: see the Lemma 3.2 in RogavaTsiklauri_LocConvg2012
  • Lemma 3.4
  • Lemma 3.5
  • Remark 3.6
  • Theorem 3.7
  • Remark 3.8
  • Remark 4.1