Numerical Modeling of Peridynamic Richards’ Equation with Piecewise Smooth Initial Conditions Using Spectral Methods

Abstract

In this paper, we introduce peridynamic theory and its application to Richards’ equation with a piecewise smooth initial condition. Peridynamic theory is a non-local continuum theory that models the deformation and failure of materials. Richards’ equation describes the unsaturated flow of water through porous media, and it plays an essential role in many applications, such as groundwater management, soil science, and environmental engineering. We develop a peridynamic formulation of Richards’ equation that includes the effect of peridynamic forces and a piecewise smooth initial condition, further introducing a non-standard symmetric influence function to describe such peridynamic interactions, which turns out to provide beneficial effects from a numerical point of view. Moreover, we implement a numerical scheme based on Chebyshev polynomials and symmetric Gauss–Lobatto nodes, providing a powerful spectral method able to capture singularities and critical issues of Richards’ equation with piecewise smooth initial conditions. We also present numerical simulations that illustrate the performance of the proposed approach. In particular, we perform a computational investigation into the spatial order of convergence, showing that, despite the discontinuity in the initial condition, the order of convergence is retained.

Iterated Crank–Nicolson Method for Peridynamic Models

“…In this example, we explore our method's capability for handling discontinuities, focusing on discontinuous initial conditions [34,35]. The simulation setup mirrors the previous linear example in Section 4.1, except for the initial condition.…”

Exploring Integral ϝ-Contractions with Applications to Integral Equations and Fractional BVPs

“…Future research on multi-point boundary value problems for the Riemann-Liouville fractional order nonlinear differential equations can focus on developing efficient numerical methods, investigating the solution existence and uniqueness, analyzing the stability properties, and exploring interdisciplinary applications. For more comprehensive information and related sources on this subject, readers are referred to [29][30][31][32], as well as the recommended sources mentioned within those references. In the cited work [33], the study revolves around the utilization of classical fixed-point theory to explore the existence of at least one solution.…”

“…where a : [x, y] → R, given functions f : [x, y] → R, and h, g : R → R, are all continuous and a parameter µ ∈ R. The integral equation's kernel, denoted by K, is defined on the domain [x, y] × [x, y]. In the specific case where both f and h correspond to the identity mapping I on R, Equation (30) is commonly referred to as a Fredholm integral equation of the second kind. Further details and relevant references on this subject can be found in [34][35][36][37], along with the cited sources therein.…”

“…and c represents the unique solution of Equation (30). Let Γ 1 and Γ 2 be self-mappings on S defined by…”