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Convergence Analysis of a Spectral Numerical Method for a Peridynamic Formulation of Richards' Equation

Fabio V. Difonzo, Sabrina F. Pellegrino

TL;DR

The paper analyzes a nonlocal peridynamic formulation of Richards' equation and develops a Chebyshev spectral spatial discretization with forward Euler time stepping. It proves the existence and uniqueness of a weak solution for the fully-discrete scheme and establishes convergence to the continuous model as Δt → 0 and N → ∞ using stability and compactness arguments. Numerical experiments with varying initial smoothness and standard constitutive relations validate the theoretical results and demonstrate robustness. The work enables high-order, nonlocal modeling of unsaturated flow and points to extensions to 2D and lower-regularity data, with potential strategies to manage Gibbs phenomena.

Abstract

We study the implementation of a Chebyshev spectral method with forward Euler integrator to investigate a peridynamic nonlocal formulation of Richards' equation. We prove the convergence of the fully-discretization of the model showing the existence and uniqueness of a solution to the weak formulation of the method by using the compactness properties of the approximated solution and exploiting the stability of the numerical scheme. We further support our results through numerical simulations, using initial conditions with different order of smoothness, showing reliability and robustness of the theoretical findings presented in the paper.

Convergence Analysis of a Spectral Numerical Method for a Peridynamic Formulation of Richards' Equation

TL;DR

The paper analyzes a nonlocal peridynamic formulation of Richards' equation and develops a Chebyshev spectral spatial discretization with forward Euler time stepping. It proves the existence and uniqueness of a weak solution for the fully-discrete scheme and establishes convergence to the continuous model as Δt → 0 and N → ∞ using stability and compactness arguments. Numerical experiments with varying initial smoothness and standard constitutive relations validate the theoretical results and demonstrate robustness. The work enables high-order, nonlocal modeling of unsaturated flow and points to extensions to 2D and lower-regularity data, with potential strategies to manage Gibbs phenomena.

Abstract

We study the implementation of a Chebyshev spectral method with forward Euler integrator to investigate a peridynamic nonlocal formulation of Richards' equation. We prove the convergence of the fully-discretization of the model showing the existence and uniqueness of a solution to the weak formulation of the method by using the compactness properties of the approximated solution and exploiting the stability of the numerical scheme. We further support our results through numerical simulations, using initial conditions with different order of smoothness, showing reliability and robustness of the theoretical findings presented in the paper.
Paper Structure (5 sections, 8 theorems, 44 equations, 3 figures)

This paper contains 5 sections, 8 theorems, 44 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A nonlocal formulation of Richards' equation based on peridynamics
  3. Fully spectral discretization of the model
  4. Numerical Simulations
  5. Conclusions

Key Result

Lemma 2.2

For any real $0\le \mu\le s$, there exists a positive constant $C$ such that

Figures (3)

  • Figure 1: The distributed influence function $\varphi_{\delta}(z)$.
  • Figure 2: With reference to Example \ref{['ex:1']}, the profile of the soil moisture for different time values. The parameters of the simulations are $N=100$, $\Delta t=0.06$ s and $\delta=0.15$.
  • Figure 3: With reference to Example \ref{['ex:2']}, the profile of the soil moisture for different time values. The parameters of the simulations are $N=100$, $\Delta t=0.06$ s and $\delta=0.15$.

Theorems & Definitions (16)

  • Remark 2.1
  • Lemma 2.2: Canuto
  • Theorem 2.3: BDFP
  • Lemma 3.1: see maxprinc2019
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 6 more