# Iterated Crank–Nicolson Method for Peridynamic Models

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Peridynamic Model

## 3. The Iterated Crank–Nicolson Method

## 4. Numerical Examples

#### 4.1. Linear Peridynamic Equation

#### 4.2. Energy Conservation for the ICN Method with Different Weights

#### 4.3. The ICN Method with Multiple Iterations

#### 4.4. Comparison of Three Quadrature Formulas for Linear Peridynamics

#### 4.5. Linear Peridynamic Model with Discontinuous Initial Condition

#### 4.6. Nonlinear Peridynamic Equation

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Silling, S.A. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids
**2000**, 48, 175–209. [Google Scholar] [CrossRef] - Silling, S.A.; Lehoucq, R.B. Peridynamic theory of solid mechanics. Adv. Appl. Mech.
**2010**, 44, 73–168. [Google Scholar] - Madenci, E.; Oterkus, E. Peridynamic Theory and Its Applications; Springer: New York, NY, USA, 2014. [Google Scholar]
- Bobaru, F.; Foster, J.T.; Geubelle, P.H.; Silling, S.A. Handbook of Peridynamic Modeling; CRC Press: Boca Raton, FL, USA, 2016. [Google Scholar]
- Javili, A.; Morasata, R.; Oterkus, E.; Oterkus, S. Peridynamics review. Math. Mech. Solids
**2019**, 24, 3714–3739. [Google Scholar] [CrossRef] - Silling, S.A.; Askari, E. A meshfree method based on the peridynamic model of solid mechanics. Comput. Struct.
**2005**, 83, 1526–1535. [Google Scholar] [CrossRef] - Weckner, O.; Emmrich, E. Numerical simulation of the dynamics of a nonlocal, inhomogeneous, infinite bar. J. Comput. Appl. Mech.
**2005**, 6, 311–319. [Google Scholar] - Emmrich, E.; Weckner, O. The peridynamic equation and its spatial discretisation. Math. Model. Anal.
**2007**, 12, 17–27. [Google Scholar] [CrossRef] - Macek, R.W.; Silling, S.A. Peridynamics via finite element analysis. Finite Elem. Anal. Des.
**2007**, 43, 1169–1178. [Google Scholar] [CrossRef] - Tian, X.; Du, Q. Nonconforming discontinuous Galerkin methods for nonlocal variational problems. SIAM J. Numer. Anal.
**2015**, 53, 762–781. [Google Scholar] [CrossRef] - Lopez, L.; Pellegrino, S.F. A spectral method with volume penalization for a nonlinear peridynamic model. Int. J. Numer. Methods Eng.
**2021**, 122, 707–725. [Google Scholar] [CrossRef] - Lopez, L.; Pellegrino, S.F. A nonperiodic Chebyshev spectral method avoiding penalization techniques for a class of nonlinear peridynamic models. Int. J. Numer. Methods Eng.
**2022**, 123, 4859–4876. [Google Scholar] [CrossRef] - Lopez, L.; Pellegrino, S.F. A space-time discretization of a nonlinear peridynamic model on a 2D lamina. Comput. Math. Appl.
**2022**, 116, 161–175. [Google Scholar] [CrossRef] - Lopez, L.; Pellegrino, S.F. A fast-convolution based space–time Chebyshev spectral method for peridynamic models. Adv. Contin. Discret. Model.
**2022**, 2022, 70. [Google Scholar] [CrossRef] - Hairer, E.; Lubich, C.; Wanner, G. Geometric numerical integration illustrated by the Störmer–Verlet method. Acta Numer.
**2003**, 12, 399–450. [Google Scholar] [CrossRef] - Seleson, P.; Pasetto, M.; John, Y.; Trageser, J.; Reeve, S.T. PDMATLAB2D: A Peridynamics MATLAB Two-dimensional Code. J. Peridyn. Nonlocal Model.
**2024**, 1–57. [Google Scholar] [CrossRef] - Coclite, G.M.; Fanizzi, A.; Lopez, L.; Maddalena, F.; Pellegrino, S.F. Numerical methods for the nonlocal wave equation of the peridynamics. Appl. Numer. Math.
**2020**, 155, 119–139. [Google Scholar] [CrossRef] - Crank, J.; Nicolson, P. A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type. In Mathematical Proceedings of the Cambridge Philosophical Society; Cambridge University Press: Cambridge, UK, 1947; Volume 43, pp. 50–67. [Google Scholar]
- Choptuik, M.W. Critical behaviour in scalar field collapse. In Deterministic Chaos in General Relativity; Springer: Boston, MA, USA, 1994; pp. 155–175. [Google Scholar]
- Teukolsky, S.A. Stability of the iterated Crank-Nicholson method in numerical relativity. Phys. Rev. D
**2000**, 61, 087501. [Google Scholar] [CrossRef] - Duez, M.D.; Marronetti, P.; Shapiro, S.L.; Baumgarte, T.W. Hydrodynamic simulations in 3 + 1 general relativity. Phys. Rev. D
**2003**, 67, 024004. [Google Scholar] [CrossRef] - Duez, M.D.; Liu, Y.T.; Shapiro, S.L.; Stephens, B.C. General relativistic hydrodynamics with viscosity: Contraction, catastrophic collapse, and disk formation in hypermassive neutron stars. Phys. Rev. D
**2004**, 69, 104030. [Google Scholar] [CrossRef] - Yioultsis, T.V.; Ziogos, G.D.; Kriezis, E.E. Explicit finite-difference vector beam propagation method based on the iterated Crank-Nicolson scheme. JOSA A
**2009**, 26, 2183–2191. [Google Scholar] [CrossRef] [PubMed] - Ketzaki, D.A.; Rekanos, I.T.; Kosmanis, T.I.; Yioultsis, T.V. Beam Propagation Method Based on the Iterated Crank–Nicolson Scheme for Solving Large-Scale Wave Propagation Problems. IEEE Trans. Magn.
**2015**, 51, 7204404. [Google Scholar] [CrossRef] - Shibayama, J.; Nishio, T.; Yamauchi, J.; Nakano, H. Explicit FDTD method based on iterated Crank–Nicolson scheme. Electron. Lett.
**2022**, 58, 16–18. [Google Scholar] [CrossRef] - Wu, P.; Wang, X.; Xie, Y.; Jiang, H.; Natsuki, T. Iterated Crank-Nicolson Procedure with Enhanced Absorption for Nonuniform Domains. IEEE J. Multiscale Multiphysics Comput. Tech.
**2022**, 7, 61–68. [Google Scholar] [CrossRef] - Leiler, G.; Rezzolla, L. Iterated Crank-Nicolson method for hyperbolic and parabolic equations in numerical relativity. Phys. Rev. D
**2006**, 73, 044001. [Google Scholar] [CrossRef] - Tran, Q.; Liu, J. Modified iterated Crank-Nicolson method with improved accuracy for advection equations. Numer. Algorithms
**2023**, 1–22. [Google Scholar] [CrossRef] - Butcher, J.C. Numerical Methods for Ordinary Differential Equations; John Wiley & Sons: Hoboken, NJ, USA, 2016. [Google Scholar]
- Burden, R.L. Numerical Analysis; Brooks/Cole Cengage Learning: Boston, MA, USA, 2011. [Google Scholar]
- Gottlieb, S.; Shu, C.W. Total variation diminishing Runge-Kutta schemes. Math. Comput.
**1998**, 67, 73–85. [Google Scholar] [CrossRef] - Gottlieb, S.; Shu, C.W.; Tadmor, E. Strong stability-preserving high-order time discretization methods. SIAM Rev.
**2001**, 43, 89–112. [Google Scholar] [CrossRef] - Weckner, O.; Abeyaratne, R. The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids
**2005**, 53, 705–728. [Google Scholar] [CrossRef] - Emmrich, E.; Weckner, O. Analysis and numerical approximation of an integro-differential equation modeling non-local effects in linear elasticity. Math. Mech. Solids
**2007**, 12, 363–384. [Google Scholar] [CrossRef] - Difonzo, F.V.; Di Lena, F. Numerical Modeling of Peridynamic Richards’ Equation with Piecewise Smooth Initial Conditions Using Spectral Methods. Symmetry
**2023**, 15, 960. [Google Scholar] [CrossRef]

**Figure 1.**Solutions to the linear peridynamic equation at (

**a**) $t=2.5$, (

**b**) $t=5$, (

**c**) $t=7.5$, and (

**d**) $t=10$. Method: the ICN method with $\theta =0.5$. Mesh size $N=800$.

**Figure 2.**Time history of the total energy variation $E\left(t\right)-E\left(0\right)$. The grid size is $N=800$.

**Figure 3.**Solutions of the linear peridynamic equation with piecewise constant initial displacement field (48) at (

**a**) $t=2.5$, (

**b**) $t=5$, (

**c**) $t=7.5$, and (

**d**) $t=10$.

**Figure 4.**Solutions of the linear peridynamic equation with piecewise constant initial velocity field (49) at (

**a**) $t=2.5$, (

**b**) $t=5$, (

**c**) $t=7.5$, and (

**d**) $t=10$.

**Figure 5.**Solutions to the nonlinear peridynamic equation at (

**a**) $t=2.5$, (

**b**) $t=5$, (

**c**) $t=7.5$, and (

**d**) $t=10$.

**Table 1.**Comparison of the SV method and ICN methods with various weights for the linear peridynamic equation. ${\u03f5}_{{L}_{2}}$ and ${\u03f5}_{{L}_{\infty}}$ represent the numerical errors measured in the ${L}_{2}$ and ${L}_{\infty}$ norms, respectively. $|{E}_{n}-{E}_{0}|$ denotes the energy variation from $t=0$ to $t=10$.

Method | N | ${\mathit{\u03f5}}_{{\mathit{L}}_{2}}$ | Rate | ${\mathit{\u03f5}}_{{\mathit{L}}_{\mathit{\infty}}}$ | Rate | $|{\mathit{E}}_{\mathit{n}}-{\mathit{E}}_{0}|$ | Rate |
---|---|---|---|---|---|---|---|

SV | 100 | 1.36 × ${10}^{-3}$ | - | 3.65 × ${10}^{-2}$ | - | 3.67 × ${10}^{-2}$ | - |

200 | 2.34 × ${10}^{-4}$ | 2.54 | 8.84 × ${10}^{-3}$ | 2.05 | 1.83 × ${10}^{-2}$ | 1.00 | |

400 | 4.10 × ${10}^{-5}$ | 2.51 | 2.19 × ${10}^{-3}$ | 2.01 | 9.18 × ${10}^{-3}$ | 1.00 | |

800 | 7.24 × ${10}^{-6}$ | 2.50 | 5.47 × ${10}^{-4}$ | 2.00 | 4.59 × ${10}^{-3}$ | 1.00 | |

1600 | 1.28 × ${10}^{-6}$ | 2.50 | 1.37 × ${10}^{-4}$ | 2.00 | 2.30 × ${10}^{-3}$ | 1.00 | |

ICN ($\theta =0.5$) | 100 | 2.34 × ${10}^{-3}$ | - | 6.18 × ${10}^{-2}$ | - | 3.87 × ${10}^{-1}$ | - |

200 | 4.67 × ${10}^{-4}$ | 2.32 | 1.76 × ${10}^{-2}$ | 1.81 | 1.45 × ${10}^{-1}$ | 1.42 | |

400 | 8.27 × ${10}^{-5}$ | 2.50 | 4.42 × ${10}^{-3}$ | 2.00 | 3.89 × ${10}^{-2}$ | 1.90 | |

800 | 1.45 × ${10}^{-5}$ | 2.51 | 1.10 × ${10}^{-3}$ | 2.01 | 9.84 × ${10}^{-3}$ | 1.98 | |

1600 | 2.56 × ${10}^{-6}$ | 2.50 | 2.74 × ${10}^{-4}$ | 2.00 | 2.47 × ${10}^{-3}$ | 2.00 | |

ICN ($\theta =\frac{1}{3}$) | 100 | 7.59 × ${10}^{-4}$ | - | 2.11 × ${10}^{-2}$ | - | 1.60 × ${10}^{-1}$ | - |

200 | 7.30 × ${10}^{-5}$ | 3.38 | 2.79 × ${10}^{-3}$ | 2.92 | 4.99 × ${10}^{-2}$ | 1.68 | |

400 | 6.57 × ${10}^{-6}$ | 3.47 | 3.56 × ${10}^{-4}$ | 2.97 | 1.30 × ${10}^{-2}$ | 1.94 | |

800 | 5.83 × ${10}^{-7}$ | 3.49 | 4.45 × ${10}^{-5}$ | 3.00 | 3.28 × ${10}^{-3}$ | 1.99 | |

1600 | 5.16 × ${10}^{-8}$ | 3.50 | 5.57 × ${10}^{-6}$ | 3.00 | 8.22 × ${10}^{-4}$ | 2.00 | |

ICN ($\theta =0.25$) | 100 | 1.45 × ${10}^{-3}$ | - | 3.91 × ${10}^{-2}$ | - | 1.58 × ${10}^{-2}$ | - |

200 | 2.38 × ${10}^{-4}$ | 2.61 | 8.99 × ${10}^{-3}$ | 2.12 | 9.70 × ${10}^{-4}$ | 4.02 | |

400 | 4.12 × ${10}^{-5}$ | 2.53 | 2.20 × ${10}^{-3}$ | 2.03 | 6.06 × ${10}^{-5}$ | 4.00 | |

800 | 7.25 × ${10}^{-6}$ | 2.51 | 5.48 × ${10}^{-4}$ | 2.00 | 3.79 × ${10}^{-6}$ | 4.00 | |

1600 | 1.28 × ${10}^{-6}$ | 2.50 | 1.37 × ${10}^{-4}$ | 2.00 | 2.37 × ${10}^{-7}$ | 4.00 |

Method | $|{\mathit{E}}_{\mathit{n}}-{\mathit{E}}_{0}|$ ($\mathit{N}=1600$) | Run Time (s) | $|{\mathit{E}}_{\mathit{n}}-{\mathit{E}}_{0}|$ ($\mathit{N}=3200$) | Run Time (s) |
---|---|---|---|---|

SV | 2.30 × ${10}^{-3}$ | 20 | 1.15 × ${10}^{-3}$ | 139 |

RK4 | 2.10 × ${10}^{-7}$ | 33 | 1.32 × ${10}^{-8}$ | 236 |

ICN ($\theta =0.25$) | 2.37 × ${10}^{-7}$ | 26 | 1.48 × ${10}^{-8}$ | 184 |

$\mathit{\theta}$ | ${\mathit{E}}_{\mathit{n}}-{\mathit{E}}_{0}$ |
---|---|

0.70 | −1.77 × ${10}^{-2}$ |

0.60 | −1.38 × ${10}^{-2}$ |

0.50 | −9.84 × ${10}^{-3}$ |

0.40 | −5.91 × ${10}^{-3}$ |

1/3 | −3.28 × ${10}^{-3}$ |

0.30 | −1.97 × ${10}^{-3}$ |

0.25 | 3.79 × ${10}^{-6}$ |

0.20 | 1.98× ${10}^{-3}$ |

**Table 4.**Energy variation of the ICN method when the number of iterations changes. ${E}_{n}$ is the total energy at $t=10$.

Number of Iterations m | ${\mathit{E}}_{\mathit{n}}-{\mathit{E}}_{0}$ ($\mathit{N}=800$) | Run Time (s) | ${\mathit{E}}_{\mathit{n}}-{\mathit{E}}_{0}$ ($\mathit{N}=1600$) | Run Time (s) |
---|---|---|---|---|

1 | 9.88 × ${10}^{-3}$ | 2.65 | 2.47 × ${10}^{-3}$ | 20.44 |

2 | −9.84 × ${10}^{-3}$ | 3.56 | −2.47 × ${10}^{-3}$ | 29.37 |

3 | −1.51 × ${10}^{-5}$ | 4.37 | −9.47 × ${10}^{-7}$ | 37.64 |

4 | 1.51 × ${10}^{-5}$ | 5.23 | 9.47 × ${10}^{-7}$ | 44.99 |

5 | 2.54 × ${10}^{-8}$ | 6.07 | 3.98 × ${10}^{-10}$ | 52.17 |

6 | −2.54 × ${10}^{-8}$ | 6.92 | −3.98 × ${10}^{-10}$ | 58.95 |

7 | −4.55 × ${10}^{-11}$ | 7.72 | −1.78 × ${10}^{-13}$ | 66.40 |

8 | 4.55 × ${10}^{-11}$ | 8.58 | 1.78 × ${10}^{-13}$ | 72.49 |

9 | 8.53 × ${10}^{-14}$ | 9.45 | ∼${10}^{-15}$ | 80.32 |

10 | −8.70 × ${10}^{-14}$ | 10.27 | ∼${10}^{-15}$ | 89.71 |

11 | ∼${10}^{-15}$ | 11.15 | ∼${10}^{-15}$ | 95.04 |

12 | ∼${10}^{-15}$ | 11.97 | ∼${10}^{-15}$ | 101.81 |

N | ${\mathit{L}}_{2}$ | Rate | ${\mathit{L}}_{\mathit{\infty}}$ | Rate | |${\mathit{E}}_{\mathit{n}}-{\mathit{E}}_{0}$| | Rate |
---|---|---|---|---|---|---|

100 | 3.11 × ${10}^{-3}$ | - | 7.78 × ${10}^{-2}$ | - | −5.22 × ${10}^{-2}$ | - |

200 | 4.90 × ${10}^{-4}$ | 2.66 | 1.86 × ${10}^{-2}$ | 2.06 | −3.77 × ${10}^{-3}$ | 3.79 |

400 | 8.31 × ${10}^{-5}$ | 2.56 | 4.44 × ${10}^{-3}$ | 2.07 | −2.41 × ${10}^{-4}$ | 3.97 |

800 | 1.45 × ${10}^{-5}$ | 2.52 | 1.10 × ${10}^{-3}$ | 2.02 | −1.51 × ${10}^{-5}$ | 3.99 |

1600 | 2.56 × ${10}^{-6}$ | 2.50 | 2.74 × ${10}^{-4}$ | 2.00 | −9.47 × ${10}^{-7}$ | 4.00 |

Number of Iterations m | Rate of Convergence of Energy |
---|---|

1 | 2.0 |

2 | 2.0 |

3 | 4.0 |

4 | 4.0 |

5 | 6.0 |

6 | 6.0 |

7 | 8.0 |

8 | 8.0 |

**Table 7.**Comparison of three quadrature formulas for the peridynamic simulation using the SV method. ${\u03f5}_{{L}_{2}}$ and ${\u03f5}_{{L}_{\infty}}$ represent the numerical errors measured in the ${L}_{2}$ and ${L}_{\infty}$ norms, respectively. $|{E}_{n}-{E}_{0}|$ denotes the energy variation from $t=0$ to $t=10$.

Method | N | ${\mathit{\u03f5}}_{{\mathit{L}}_{2}}$ | Rate | ${\mathit{\u03f5}}_{{\mathit{L}}_{\mathit{\infty}}}$ | Rate | $|{\mathit{E}}_{\mathit{n}}-{\mathit{E}}_{0}|$ | Rate |
---|---|---|---|---|---|---|---|

SV + midpoint | 100 | 1.3626781938 × ${10}^{-3}$ | - | 3.6546374629 × ${10}^{-2}$ | - | −3.6686416132 × ${10}^{-2}$ | - |

200 | 2.3386240754 × ${10}^{-4}$ | 2.54 | 8.8420752239 × ${10}^{-3}$ | 2.05 | −1.8341493309 × ${10}^{-2}$ | 1.00 | |

400 | 4.1038150512 × ${10}^{-5}$ | 2.51 | 2.1896453314 × ${10}^{-3}$ | 2.01 | −9.1848205222 × ${10}^{-3}$ | 1.00 | |

800 | 7.2412556153 × ${10}^{-6}$ | 2.50 | 5.4733686142 × ${10}^{-4}$ | 2.00 | −4.5945520876 × ${10}^{-3}$ | 1.00 | |

1600 | 1.2794967586 × ${10}^{-6}$ | 2.50 | 1.3681318781 × ${10}^{-4}$ | 2.00 | −2.2975549644 × ${10}^{-3}$ | 1.00 | |

SV + composite trapezoidal | 100 | 1.3626781938 × ${10}^{-3}$ | - | 3.6546374631 × ${10}^{-2}$ | - | −3.6686416132 × ${10}^{-2}$ | - |

200 | 2.3386240741 × ${10}^{-4}$ | 2.54 | 8.8420752309 × ${10}^{-3}$ | 2.05 | −1.8341493309 × ${10}^{-2}$ | 1.00 | |

400 | 4.1038150469 × ${10}^{-5}$ | 2.51 | 2.1896453350 × ${10}^{-3}$ | 2.01 | −9.1848205222 × ${10}^{-3}$ | 1.00 | |

800 | 7.2412556001 × ${10}^{-6}$ | 2.50 | 5.4733686322 × ${10}^{-4}$ | 2.00 | −4.5945520876 × ${10}^{-3}$ | 1.00 | |

1600 | 1.2794967533 × ${10}^{-6}$ | 2.50 | 1.3681318870 × ${10}^{-4}$ | 2.00 | −2.2975549644 × ${10}^{-3}$ | 1.00 | |

SV + composite Simpson | 100 | 1.3532560898 × ${10}^{-3}$ | - | 3.6461913740 × ${10}^{-2}$ | - | −3.6637130961 × ${10}^{-2}$ | - |

200 | 2.3386240745 × ${10}^{-4}$ | 2.53 | 8.8420752291 × ${10}^{-3}$ | 2.04 | −1.8341493309 × ${10}^{-2}$ | 1.00 | |

400 | 4.1038150476 × ${10}^{-5}$ | 2.51 | 2.1896453345 × ${10}^{-3}$ | 2.01 | −9.1848205222 × ${10}^{-3}$ | 1.00 | |

800 | 7.2412556013 × ${10}^{-6}$ | 2.50 | 5.4733686308 × ${10}^{-4}$ | 2.00 | −4.5945520876 × ${10}^{-3}$ | 1.00 | |

1600 | 1.2794967535 × ${10}^{-6}$ | 2.50 | 1.3681318867 × ${10}^{-4}$ | 2.00 | −2.2975549644 × ${10}^{-3}$ | 1.00 |

**Table 8.**Comparison of three quadrature formulas for the peridynamic simulation using the ICN method ($\theta =0.25$). ${\u03f5}_{{L}_{2}}$ and ${\u03f5}_{{L}_{\infty}}$ represent the numerical errors measured in the ${L}_{2}$ and ${L}_{\infty}$ norms, respectively. $|{E}_{n}-{E}_{0}|$ denotes the energy variation from $t=0$ to $t=10$.

Method | N | ${\mathit{\u03f5}}_{{\mathit{L}}_{2}}$ | Rate | ${\mathit{\u03f5}}_{{\mathit{L}}_{\mathit{\infty}}}$ | Rate | $|{\mathit{E}}_{\mathit{n}}-{\mathit{E}}_{0}|$ | Rate |
---|---|---|---|---|---|---|---|

ICN + midpoint | 100 | 1.4531942146 × ${10}^{-3}$ | - | 3.9071358030 × ${10}^{-2}$ | - | 1.5796799025 × ${10}^{-2}$ | - |

200 | 2.3797519266 × ${10}^{-4}$ | 2.61 | 8.9850019534 × ${10}^{-3}$ | 2.12 | 9.7041252646 × ${10}^{-4}$ | 4.02 | |

400 | 4.1222863495 × ${10}^{-5}$ | 2.53 | 2.1984031675 × ${10}^{-3}$ | 2.03 | 6.0617629614 × ${10}^{-5}$ | 4.00 | |

800 | 7.2494508340 × ${10}^{-6}$ | 2.51 | 5.4789636190 × ${10}^{-4}$ | 2.00 | 3.7885364605 × ${10}^{-6}$ | 4.00 | |

1600 | 1.2798591070 × ${10}^{-6}$ | 2.50 | 1.3684841395 × ${10}^{-4}$ | 2.00 | 2.3678339289 × ${10}^{-7}$ | 4.00 | |

ICN + composite trapezoidal | 100 | 1.4531942145 × ${10}^{-3}$ | - | 3.9071358029 × ${10}^{-2}$ | - | 1.5796799025 × ${10}^{-2}$ | - |

200 | 2.3797519254 × ${10}^{-4}$ | 2.61 | 8.9850019605 × ${10}^{-3}$ | 2.12 | 9.7041252645 × ${10}^{-4}$ | 4.02 | |

400 | 4.1222863452 × ${10}^{-5}$ | 2.53 | 2.1984031712 × ${10}^{-3}$ | 2.03 | 6.0617629611 × ${10}^{-5}$ | 4.00 | |

800 | 7.2494508188 × ${10}^{-6}$ | 2.51 | 5.4789636369 × ${10}^{-4}$ | 2.00 | 3.7885364605 × ${10}^{-6}$ | 4.00 | |

1600 | 1.2798591017 × ${10}^{-6}$ | 2.50 | 1.3684841484 × ${10}^{-4}$ | 2.00 | 2.3678340710 × ${10}^{-7}$ | 4.00 | |

ICN + composite Simpson | 100 | 1.4433953589 × ${10}^{-3}$ | - | 3.8867087321 × ${10}^{-2}$ | - | 1.5770288980 × ${10}^{-2}$ | - |

200 | 2.3797519257 × ${10}^{-4}$ | 2.60 | 8.9850019587 × ${10}^{-3}$ | 2.11 | 9.7041252645 × ${10}^{-4}$ | 4.02 | |

400 | 4.1222863459 × ${10}^{-5}$ | 2.53 | 2.1984031706 × ${10}^{-3}$ | 2.03 | 6.0617629612 × ${10}^{-5}$ | 4.00 | |

800 | 7.2494508201 × ${10}^{-6}$ | 2.51 | 5.4789636355 × ${10}^{-4}$ | 2.00 | 3.7885364570 × ${10}^{-6}$ | 4.00 | |

1600 | 1.2798591019 × ${10}^{-6}$ | 2.50 | 1.3684841481 × ${10}^{-4}$ | 2.00 | 2.3678339645 × ${10}^{-7}$ | 4.00 |

**Table 9.**Numerical errors and convergence rates of the SV and the ICN methods for the linear peridynamic equation with piecewise constant initial displacement field (48). ${\u03f5}_{{L}_{2}}$ and ${\u03f5}_{{L}_{\infty}}$ represent the numerical errors measured in the ${L}_{2}$ and ${L}_{\infty}$ norms, respectively. $|{E}_{n}-{E}_{0}|$ denotes the energy variation from $t=0$ to $t=10$.

Method | N | ${\mathit{\u03f5}}_{{\mathit{L}}_{2}}$ | Rate | ${\mathit{\u03f5}}_{{\mathit{L}}_{\mathit{\infty}}}$ | Rate | $|{\mathit{E}}_{\mathit{n}}-{\mathit{E}}_{0}|$ | Rate |
---|---|---|---|---|---|---|---|

SV | 100 | 3.31 × ${10}^{-3}$ | - | 1.03 × ${10}^{-1}$ | - | 6.40 × ${10}^{-2}$ | - |

200 | 4.78 × ${10}^{-4}$ | 2.79 | 2.74 × ${10}^{-2}$ | 1.91 | 3.26 × ${10}^{-2}$ | 0.97 | |

400 | 8.17 × ${10}^{-5}$ | 2.55 | 7.53 × ${10}^{-3}$ | 1.86 | 1.63 × ${10}^{-2}$ | 1.00 | |

800 | 1.36 × ${10}^{-5}$ | 2.58 | 1.90 × ${10}^{-3}$ | 1.99 | 8.14 × ${10}^{-3}$ | 1.00 | |

1600 | 1.89 × ${10}^{-6}$ | 2.85 | 3.91 × ${10}^{-4}$ | 2.28 | 4.06 × ${10}^{-3}$ | 1.00 | |

ICN ($\theta =0.25$) | 100 | 3.53 × ${10}^{-3}$ | - | 1.09 × ${10}^{-1}$ | - | 3.13 × ${10}^{-2}$ | - |

200 | 4.89 × ${10}^{-4}$ | 2.85 | 2.81 × ${10}^{-2}$ | 1.95 | 2.00 × ${10}^{-3}$ | 3.96 | |

400 | 8.22 × ${10}^{-5}$ | 2.57 | 7.59 × ${10}^{-3}$ | 1.89 | 1.26 × ${10}^{-4}$ | 3.99 | |

800 | 1.37 × ${10}^{-5}$ | 2.59 | 1.90 × ${10}^{-3}$ | 2.00 | 7.88 × ${10}^{-6}$ | 4.00 | |

1600 | 1.89 × ${10}^{-6}$ | 2.85 | 3.91 × ${10}^{-4}$ | 2.28 | 4.92 × ${10}^{-7}$ | 4.00 |

**Table 10.**Numerical errors and convergence rates of the SV and the ICN methods for the linear peridynamic equation with piecewise constant initial velocity field (49). ${\u03f5}_{{L}_{2}}$ and ${\u03f5}_{{L}_{\infty}}$ represent the numerical errors measured in the ${L}_{2}$ and ${L}_{\infty}$ norms, respectively. $|{E}_{n}-{E}_{0}|$ denotes the energy variation from $t=0$ to $t=10$.

Method | N | ${\mathit{\u03f5}}_{{\mathit{L}}_{2}}$ | Rate | ${\mathit{\u03f5}}_{{\mathit{L}}_{\mathit{\infty}}}$ | Rate | $|{\mathit{E}}_{\mathit{n}}-{\mathit{E}}_{0}|$ | Rate |
---|---|---|---|---|---|---|---|

SV | 100 | 3.47 × ${10}^{-3}$ | - | 8.95 × ${10}^{-2}$ | - | 2.99 × ${10}^{-2}$ | - |

200 | 2.69 × ${10}^{-4}$ | 3.69 | 9.93 × ${10}^{-3}$ | 3.17 | 1.51 × ${10}^{-2}$ | 0.99 | |

400 | 4.69 × ${10}^{-5}$ | 2.52 | 2.61 × ${10}^{-3}$ | 1.93 | 7.54 × ${10}^{-3}$ | 1.00 | |

800 | 7.93 × ${10}^{-6}$ | 2.56 | 6.62 × ${10}^{-4}$ | 1.98 | 3.76 × ${10}^{-3}$ | 1.00 | |

1600 | 1.15 × ${10}^{-6}$ | 2.78 | 1.41 × ${10}^{-4}$ | 2.23 | 1.88 × ${10}^{-3}$ | 1.00 | |

ICN ($\theta =0.25$) | 100 | 3.50 × ${10}^{-3}$ | - | 9.14 × ${10}^{-2}$ | - | 1.04 × ${10}^{-2}$ | - |

200 | 2.64 × ${10}^{-4}$ | 3.73 | 9.78 × ${10}^{-3}$ | 3.22 | 6.35 × ${10}^{-4}$ | 4.03 | |

400 | 4.55 × ${10}^{-5}$ | 2.54 | 2.38 × ${10}^{-3}$ | 2.04 | 3.94 × ${10}^{-5}$ | 4.01 | |

800 | 7.67 × ${10}^{-6}$ | 2.57 | 5.81 × ${10}^{-4}$ | 2.04 | 2.45 × ${10}^{-6}$ | 4.01 | |

1600 | 1.11 × ${10}^{-6}$ | 2.78 | 1.25 × ${10}^{-4}$ | 2.22 | 1.53 × ${10}^{-7}$ | 4.00 |

**Table 11.**Comparison of the SV method and the ICN methods with various weights for the nonlinear peridynamic equation. ${\u03f5}_{{L}_{2}}$ and ${\u03f5}_{{L}_{\infty}}$ represent the numerical errors measured in the ${L}_{2}$ and ${L}_{\infty}$ norms, respectively.

Method | N | ${\mathit{\u03f5}}_{{\mathit{L}}_{2}}$ | Rate | ${\mathit{\u03f5}}_{{\mathit{L}}_{\mathit{\infty}}}$ | Rate |
---|---|---|---|---|---|

SV | 100 | 2.12 × ${10}^{-4}$ | - | 4.45 × ${10}^{-3}$ | - |

200 | 2.33 × ${10}^{-5}$ | 3.19 | 7.04 × ${10}^{-4}$ | 2.66 | |

400 | 4.11 × ${10}^{-6}$ | 2.50 | 1.91 × ${10}^{-4}$ | 1.88 | |

800 | 7.65 × ${10}^{-7}$ | 2.42 | 6.45 × ${10}^{-5}$ | 1.57 | |

ICN ($\theta =0.5$) | 100 | 2.15 × ${10}^{-4}$ | - | 7.65 × ${10}^{-2}$ | - |

200 | 2.46 × ${10}^{-5}$ | 3.13 | 1.54 × ${10}^{-2}$ | 2.31 | |

400 | 4.20 × ${10}^{-6}$ | 2.55 | 3.01 × ${10}^{-3}$ | 2.36 | |

800 | 7.68 × ${10}^{-7}$ | 2.45 | 6.26 × ${10}^{-4}$ | 2.27 | |

ICN ($\theta =\frac{1}{3}$) | 100 | 8.39 × ${10}^{-4}$ | - | 2.14 × ${10}^{-2}$ | - |

200 | 8.03 × ${10}^{-5}$ | 3.39 | 2.84 × ${10}^{-3}$ | 2.92 | |

400 | 7.36 × ${10}^{-6}$ | 3.45 | 3.63 × ${10}^{-4}$ | 2.97 | |

800 | 6.59 × ${10}^{-7}$ | 3.48 | 5.17 × ${10}^{-5}$ | 2.81 | |

ICN ($\theta =0.25$) | 100 | 3.11 × ${10}^{-4}$ | - | 8.13 × ${10}^{-3}$ | - |

200 | 3.34 × ${10}^{-5}$ | 3.22 | 1.16 × ${10}^{-3}$ | 2.81 | |

400 | 4.77 × ${10}^{-6}$ | 2.81 | 2.40 × ${10}^{-4}$ | 2.27 | |

800 | 7.92 × ${10}^{-7}$ | 2.59 | 7.16 × ${10}^{-5}$ | 1.75 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Liu, J.; Appiah-Adjei, S.; Brio, M.
Iterated Crank–Nicolson Method for Peridynamic Models. *Dynamics* **2024**, *4*, 192-207.
https://doi.org/10.3390/dynamics4010011

**AMA Style**

Liu J, Appiah-Adjei S, Brio M.
Iterated Crank–Nicolson Method for Peridynamic Models. *Dynamics*. 2024; 4(1):192-207.
https://doi.org/10.3390/dynamics4010011

**Chicago/Turabian Style**

Liu, Jinjie, Samuel Appiah-Adjei, and Moysey Brio.
2024. "Iterated Crank–Nicolson Method for Peridynamic Models" *Dynamics* 4, no. 1: 192-207.
https://doi.org/10.3390/dynamics4010011