Second-Order Iterative Time Integration Scheme for Linear Poroelasticity

Core Concepts

Proposing a second-order iterative time integration scheme for linear poroelasticity.

Abstract

The content introduces a novel time stepping method for linear poroelasticity, focusing on second-order convergence. It discusses the decoupling of equations, stability concerns, and iterative approaches. The convergence analysis is detailed with proofs and assumptions. Directory: Abstract Proposes a novel time stepping method for linear poroelasticity. Introduction Discusses the application of poroelasticity in various fields. Preliminaries Introduces the equations of linear poroelasticity. Time Stepping Schemes of Second Order Presents implicit and semi-explicit discretization schemes. Second-Order Fixed Stress Explores an iterative method based on fixed stress. A First Iterative Approach Introduces an iterative scheme with first-order convergence. A Matrix Splitting for the Increments Rewrites schemes in terms of increments for iterative purposes. Novel Iterative Second-Order Scheme Details an iterative scheme with fixed inner iteration steps. Convergence Analysis Proves second-order convergence of the proposed scheme.

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Date: March 20, 2024

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Key Insights Distilled From

A second-order iterative time integration scheme for linear poroelasticity

by R. Altmann,M... at arxiv.org 03-20-2024

https://arxiv.org/pdf/2403.12699.pdf

Deeper Inquiries

How does the proposed second-order scheme compare to existing methods

The proposed second-order iterative scheme in poroelasticity offers several advantages compared to existing methods. Firstly, it extends the capabilities of traditional decoupling schemes by achieving second-order convergence. This higher order of accuracy leads to more precise numerical solutions and reduces the error associated with approximations. Additionally, the scheme allows for a fixed number of inner iteration steps, providing better control over computational resources and efficiency.

What are the practical implications of achieving second-order convergence in poroelasticity simulations

Achieving second-order convergence in poroelasticity simulations has significant practical implications. It enhances the accuracy and reliability of simulation results, making them more closely aligned with real-world behavior. This is crucial for applications in geomechanics, biological tissue modeling, and other fields where accurate predictions are essential for decision-making processes. Improved convergence properties also enable researchers to study complex phenomena with greater confidence in their numerical models.

How might variations in material parameters affect the stability and convergence properties of the iterative scheme

Variations in material parameters can have a notable impact on the stability and convergence properties of the iterative scheme proposed for poroelasticity simulations. Specifically, changes in parameters like coupling strength (ω), stiffness matrices (A, B, C), or time step size (τ) can influence the overall behavior of the iterative method. Higher values of ω may lead to instability unless appropriate relaxation techniques are employed within the scheme. Adjusting material parameters requires careful consideration to maintain stability while ensuring efficient convergence towards accurate solutions.