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Numerical analysis of a finite volume method for a 1-D wave equation with non smooth wave speed and localized Kelvin-Voigt damping

Stéphane Gerbi, Rayan Nasser, Ali Wehbe

TL;DR

The paper develops and analyzes Finite Volume Method discretizations (explicit and semi-implicit) for a 1-D elastic/viscoelastic transmission problem with a discontinuous wave speed and localized Kelvin-Voigt damping. It establishes discrete energy dissipation, derives stability bounds, and proves convergence of the numerical solution to the continuous weak solution. Numerical experiments confirm energy conservation in the undamped case and energy decay under damping, with decay profiles ranging from exponential to polynomial depending on propagation speeds and damping placement. The work fills a gap in the numerical analysis of transmission problems with interface discontinuities by providing a rigorous discrete-energy framework and convergence results that corroborate the theoretical continuous-energy decay described in prior literature.

Abstract

In this paper, we study the numerical solution of an elastic/viscoelastic wave equation with non smooth wave speed and internal localized distributed Kelvin-Voigt damping acting faraway from the boundary. Our method is based on the Finite Volume Method (FVM) and we are interested in deriving the stability estimates and the convergence of the numerical solution to the continuous one. Numerical experiments are performed to confirm the theoretical study on the decay rate of the solution to the null one when a localized damping acts.

Numerical analysis of a finite volume method for a 1-D wave equation with non smooth wave speed and localized Kelvin-Voigt damping

TL;DR

The paper develops and analyzes Finite Volume Method discretizations (explicit and semi-implicit) for a 1-D elastic/viscoelastic transmission problem with a discontinuous wave speed and localized Kelvin-Voigt damping. It establishes discrete energy dissipation, derives stability bounds, and proves convergence of the numerical solution to the continuous weak solution. Numerical experiments confirm energy conservation in the undamped case and energy decay under damping, with decay profiles ranging from exponential to polynomial depending on propagation speeds and damping placement. The work fills a gap in the numerical analysis of transmission problems with interface discontinuities by providing a rigorous discrete-energy framework and convergence results that corroborate the theoretical continuous-energy decay described in prior literature.

Abstract

In this paper, we study the numerical solution of an elastic/viscoelastic wave equation with non smooth wave speed and internal localized distributed Kelvin-Voigt damping acting faraway from the boundary. Our method is based on the Finite Volume Method (FVM) and we are interested in deriving the stability estimates and the convergence of the numerical solution to the continuous one. Numerical experiments are performed to confirm the theoretical study on the decay rate of the solution to the null one when a localized damping acts.

Paper Structure

This paper contains 13 sections, 8 theorems, 196 equations, 34 figures.

Table of Contents

  1. Introduction
  2. Construction of the discretization of the problem
  3. Admissible one-dimensional mesh and defintition of the discrete unknowns
  4. Construction of the explicit discretized problem.
  5. Dissipation of the discrete energy for the explicit discretized problem
  6. Construction of the implicit discretized problem.
  7. Dissipation of the discrete energy for the implicit discretized problem
  8. Stability estimates
  9. Convergence
  10. Numerical Experiments: Validation of the theoretical results
  11. Equal speed of propagation
  12. Different speed of propagation
  13. Conclusions

Key Result

Theorem 1

The total discrete energy defined by discrete-energy of the explicit numerical scheme Discrete Explicit satisfies the following dissipativity estimation

Figures (34)

  • Figure 1: Partial viscoelastic material
  • Figure 2: A model representing the admissible one-dimensional mesh
  • Figure 3: Initial profile
  • Figure 4: No damping/equal speed
  • Figure 5: No damping/different speed
  • ...and 29 more figures

Theorems & Definitions (24)

  • Definition 1: Weak solution
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Remark 4
  • Theorem 2
  • proof
  • Definition 2: Discrete norms
  • ...and 14 more