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# Finite Difference-Peridynamic Differential Operator for Solving Transient Heat Conduction Problems

Chunlei Ruan1,2,*, Cengceng Dong1, Zeyue Zhang1, Boyu Chen1, Zhijun Liu1

1 School of Mathematics & Statistics, Henan University of Science & Technology, Luoyang, 471023, China
2 Longmen Laboratory, Luoyang, 471023, China

* Corresponding Author: Chunlei Ruan. Email:

(This article belongs to the Special Issue: New Trends on Meshless Method and Numerical Analysis)

Computer Modeling in Engineering & Sciences 2024, 140(3), 2707-2728. https://doi.org/10.32604/cmes.2024.050003

## Abstract

Transient heat conduction problems widely exist in engineering. In previous work on the peridynamic differential operator (PDDO) method for solving such problems, both time and spatial derivatives were discretized using the PDDO method, resulting in increased complexity and programming difficulty. In this work, the forward difference formula, the backward difference formula, and the centered difference formula are used to discretize the time derivative, while the PDDO method is used to discretize the spatial derivative. Three new schemes for solving transient heat conduction equations have been developed, namely, the forward-in-time and PDDO in space (FT-PDDO) scheme, the backward-in-time and PDDO in space (BT-PDDO) scheme, and the central-in-time and PDDO in space (CT-PDDO) scheme. The stability and convergence of these schemes are analyzed using the Fourier method and Taylor’s theorem. Results show that the FT-PDDO scheme is conditionally stable, whereas the BT-PDDO and CT-PDDO schemes are unconditionally stable. The stability conditions for the FT-PDDO scheme are less stringent than those of the explicit finite element method and explicit finite difference method. The convergence rate in space for these three methods is two. These constructed schemes are applied to solve one-dimensional and two-dimensional transient heat conduction problems. The accuracy and validity of the schemes are verified by comparison with analytical solutions.

## Keywords

APA Style
Ruan, C., Dong, C., Zhang, Z., Chen, B., Liu, Z. (2024). Finite difference-peridynamic differential operator for solving transient heat conduction problems. Computer Modeling in Engineering & Sciences, 140(3), 2707-2728. https://doi.org/10.32604/cmes.2024.050003
Vancouver Style
Ruan C, Dong C, Zhang Z, Chen B, Liu Z. Finite difference-peridynamic differential operator for solving transient heat conduction problems. Comput Model Eng Sci. 2024;140(3):2707-2728 https://doi.org/10.32604/cmes.2024.050003
IEEE Style
C. Ruan, C. Dong, Z. Zhang, B. Chen, and Z. Liu "Finite Difference-Peridynamic Differential Operator for Solving Transient Heat Conduction Problems," Comput. Model. Eng. Sci., vol. 140, no. 3, pp. 2707-2728. 2024. https://doi.org/10.32604/cmes.2024.050003

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