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An unconditionally stable numerical approach for solving a nonlinear distributed delay Sobolev model

Eric Ngondiep

TL;DR

This work addresses nonlinear Sobolev equations with distributed delay by introducing an unconditionally stable, high‑order numerical scheme that couples a time-derivative interpolation with a finite element spatial discretization. The method achieves spatial fourth‑order accuracy and temporal second‑order convergence and is analyzed in a strong norm equivalent to $H^{1}$, with stability proven unconditionally. The authors provide rigorous stability and error estimates and validate them through 1D and 2D numerical experiments that also demonstrate superior efficiency over existing approaches for delay Sobolev problems. The proposed approach offers a practical, implementable tool for delay memory PDEs with potential extensions to distributed‑order or fractional time models.

Abstract

This paper proposes an unconditionally stable numerical method for solving a nonlinear Sobolev model with distributed delay. The proposed computational approach approximates the time derivative by interpolation technique whereas the spatial derivatives are approximated using the finite element approximation. This combination is simple and easy to implement. Both stability and error estimates of the constructed method are deeply analyzed in a strong norm which is equivalent to the $H^{1}$-norm. The theoretical results indicate that the constructed approach is unconditionally stable, spatial fourth-order accurate, second-order convergent in time and more efficient than a large class of numerical methods discussed in the literature for solving a general class of delay Sobolev problems. Some numerical examples are carried out to confirm the theory and demonstrate the applicability and validity of the developed technique.

An unconditionally stable numerical approach for solving a nonlinear distributed delay Sobolev model

TL;DR

This work addresses nonlinear Sobolev equations with distributed delay by introducing an unconditionally stable, high‑order numerical scheme that couples a time-derivative interpolation with a finite element spatial discretization. The method achieves spatial fourth‑order accuracy and temporal second‑order convergence and is analyzed in a strong norm equivalent to , with stability proven unconditionally. The authors provide rigorous stability and error estimates and validate them through 1D and 2D numerical experiments that also demonstrate superior efficiency over existing approaches for delay Sobolev problems. The proposed approach offers a practical, implementable tool for delay memory PDEs with potential extensions to distributed‑order or fractional time models.

Abstract

This paper proposes an unconditionally stable numerical method for solving a nonlinear Sobolev model with distributed delay. The proposed computational approach approximates the time derivative by interpolation technique whereas the spatial derivatives are approximated using the finite element approximation. This combination is simple and easy to implement. Both stability and error estimates of the constructed method are deeply analyzed in a strong norm which is equivalent to the -norm. The theoretical results indicate that the constructed approach is unconditionally stable, spatial fourth-order accurate, second-order convergent in time and more efficient than a large class of numerical methods discussed in the literature for solving a general class of delay Sobolev problems. Some numerical examples are carried out to confirm the theory and demonstrate the applicability and validity of the developed technique.

Paper Structure

This paper contains 5 sections, 3 theorems, 131 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Description of the proposed computational technique
  3. Analysis of stability and error estimates
  4. Numerical experiments
  5. General conclusions and future works

Key Result

Lemma 3.1

The bilinear operator $A(\cdot,\cdot)$ defined by equation $(17)$ is symmetric and positive definite on $V\times V$.

Figures (2)

  • Figure 1: exact solution, numerical one and error, corresponding to Example 1
  • Figure 2: analytical solution, approximate one and error, associated with Example 2

Theorems & Definitions (6)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.1
  • proof