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Crank-Nicolson-type iterative decoupled algorithms for Biot's consolidation model using total pressure

Huipeng Gu, Mingchao Cai, Jingzhi Li

Abstract

In this work, we develop Crank-Nicolson-type iterative decoupled algorithms for a three-field formulation of Biot's consolidation model using total pressure. We begin by constructing an equivalent fully implicit coupled algorithm using the standard Crank-Nicolson method for the three-field formulation of Biot's model. Employing an iterative decoupled scheme to decompose the resulting coupled system, we derive two distinctive forms of Crank-Nicolson-type iterative decoupled algorithms based on the order of temporal computation and iteration: a time-stepping iterative decoupled algorithm and a global-in-time iterative decoupled algorithm. Notably, the proposed global-in-time algorithm supports a partially parallel-in-time feature. Capitalizing on the convergence properties of the iterative decoupled scheme, both algorithms exhibit second-order time accuracy and unconditional stability. Through numerical experiments, we validate theoretical predictions and demonstrate the effectiveness and efficiency of these novel approaches.

Crank-Nicolson-type iterative decoupled algorithms for Biot's consolidation model using total pressure

Abstract

In this work, we develop Crank-Nicolson-type iterative decoupled algorithms for a three-field formulation of Biot's consolidation model using total pressure. We begin by constructing an equivalent fully implicit coupled algorithm using the standard Crank-Nicolson method for the three-field formulation of Biot's model. Employing an iterative decoupled scheme to decompose the resulting coupled system, we derive two distinctive forms of Crank-Nicolson-type iterative decoupled algorithms based on the order of temporal computation and iteration: a time-stepping iterative decoupled algorithm and a global-in-time iterative decoupled algorithm. Notably, the proposed global-in-time algorithm supports a partially parallel-in-time feature. Capitalizing on the convergence properties of the iterative decoupled scheme, both algorithms exhibit second-order time accuracy and unconditional stability. Through numerical experiments, we validate theoretical predictions and demonstrate the effectiveness and efficiency of these novel approaches.
Paper Structure (15 sections, 5 theorems, 81 equations, 7 figures, 2 tables, 3 algorithms)

This paper contains 15 sections, 5 theorems, 81 equations, 7 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. The weak form and coupled numerical schemes
  3. Preliminaries
  4. Semi-discrete numerical scheme
  5. Crank-Nicolson-type fully coupled algorithms
  6. Crank-Nicolson-type iterative decoupled algorithms
  7. A semi-discrete iterative decoupled algorithm
  8. A time-stepping iterative decoupled algorithm
  9. A global-in-time iterative decoupled algorithm
  10. Numerical experiments
  11. Example 1: the convergence behavior of Algorithms \ref{['algo:Coupled_CN']}, \ref{['algo:TIDA_CN']}, \ref{['algo:GIDA_CN']}.
  12. Example 2: Barry-Mercer's model
  13. Example 3: Mandel's problem
  14. Conclusions
  15. Acknowledgements

Key Result

Theorem 2.2

Let $(\bm{u}^N,\xi^N,p^N)$ and $(\bm{u}^N_h,\xi^N_h,p^N_h)$ be solutions of problem c1-c3 and problem d1-d3 at the final time $t_N = T$, respectively. Under Assumption assump, there holds

Figures (7)

  • Figure 1: The connections among solution sequences produced by the iterative decoupled algorithms, the numerical solution derived from the reformulated Crank-Nicolson coupled algorithm, and the exact solution.
  • Figure 2: Progression in the time-stepping iterative decoupled algorithm.
  • Figure 3: Progression in the global-in-time iterative decoupled algorithm.
  • Figure 4: Convergence behaviors of Algorithm \ref{['algo:TIDA_CN']} and Algorithm \ref{['algo:GIDA_CN']} for Example 1.
  • Figure 5: Convergence behaviors of Algorithm \ref{['algo:TIDA_CN']} and Algorithm \ref{['algo:GIDA_CN']} for Example 2.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 2.2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.3
  • Theorem 3.4
  • proof
  • Proposition 3.5: Speedup of the global-in-time algorithm