Enhancing Student Understanding of Heat Distribution in Metal Rods Through Interactive Learning Using Finite Difference Methods

Authors

  • Galih Dwi Satrio Departemen of Physics Education, Universitas Negeri Jakarta Jl. Ramawangun Muka, Jakarta 13220, Indonesia
  • Nadia Puspa Dewi Departemen of Physics Education, Universitas Negeri Jakarta Jl. Ramawangun Muka, Jakarta 13220, Indonesia
  • Rahil Nurul Fadilah Departemen of Physics Education, Universitas Negeri Jakarta Jl. Ramawangun Muka, Jakarta 13220, Indonesia
  • Zahra Dinda Palupi Departemen of Physics Education, Universitas Negeri Jakarta Jl. Ramawangun Muka, Jakarta 13220, Indonesia

DOI:

https://doi.org/10.58797/cser.020102

Keywords:

crank-nicolson, diffusion equation, explicit method, finite difference, implicit method, thermal diffusion

Abstract

This study discusses the comparison of two finite difference methods, namely the explicit method and the Crank-Nicolson method (implicit), in simulating heat propagation in a metal rod. Heating is done by lighting a candle under the metal rod which is then extinguished after some time. This research aims to improve students' understanding of heat distribution in metal rods through an interactive method based on the Finite Difference Method, which is also expected to improve the ability to analyze and apply physics concepts in a practical context. The simulation results show that the explicit method requires very small time steps to achieve good stability, resulting in longer computation times. On the other hand, the Crank-Nicolson method demonstrates better and more consistent numerical stability, even with larger time intervals. Experimental modifications with varying time intervals show that the Crank-Nicolson method remains stable and provides more accurate results compared to the explicit method. Therefore, the Crank-Nicolson method is more recommended for long-term simulations requiring high stability and accuracy.

References

Chapra, S. C., & Canale, R. P. (2015). Numerical Methods for Engineers. McGraw-Hill Education.

Dowling, N., Floerchinger, S., & Haas, T. (2020). Second law of thermodynamics for relativistic fluids formulated with relative entropy. Physical Review. D/Physical Review. D., 102(10). https://doi.org/10.1103/physrevd.102.105002

Farlow, S. J. (1994). An Introduction to Differential Equations and Their Applications. Courier Corporation.

Harewood, F. J., & McHugh, P. E. (2007). Comparison of the implicit and explicit finite element methods using crystal plasticity. Computational Materials Science, 39(2), 481–494. https://doi.org/10.1016/j.commatsci.2006.08.002

Johnson, O. B., & Oluwaseun, L. I. (2020). Crank-Nicolson and Modified Crank-Nicolson Scheme for One Dimensional Parabolic Equation. International Journal of Applied Mathematics and Theoretical Physics, 6(3), 35. https://doi.org/10.11648/j.ijamtp.20200603.11

Jung, J., Kim, H., Shin, H., & Choi, M. (2024). CEENs: Causality-enforced evolutional networks for solving time-dependent partial differential equations. Computer Methods in Applied Mechanics and Engineering, 427, 117036–117036. https://doi.org/10.1016/j.cma.2024.117036

Kalogeris, I., & Papadopoulos, V. (2021). Diffusion maps-aided Neural Networks for the solution of parametrized PDEs. Computer Methods in Applied Mechanics and Engineering, 376, 113568. https://doi.org/10.1016/j.cma.2020.113568

Lang, J., & Schmitt, B. A. (2023). Exact Discrete Solutions of Boundary Control Problems for the 1D Heat Equation. Journal of Optimization Theory and Applications, 196(3), 1106–1118. https://doi.org/10.1007/s10957-022-02154-4

Li, W., & Carvalho, R. (2024). Automating the Discovery of Partial Differential Equations in Dynamical Systems. Machine Learning Science and Technology. https://doi.org/10.1088/2632-2153/ad682f

Mitropoulos, T., Bairaktarova, D., & Huxtable, S. (2023). The utility of mechanical objects: Aiding students’ learning of abstract and difficult engineering concepts. Journal of Engineering Education, 113(1). https://doi.org/10.1002/jee.20573

Msmali, A. H., Tamsir, M., & Ahmadini, A. H. (2021). Crank-Nicolson-DQM based on cubic exponential B-splines for the approximation of nonlinear Sine-Gordon equation. Ain Shams Engineering Journal, 12(4), 4091–4097. https://doi.org/10.1016/j.asej.2021.04.004

Sanjaya, F., & Mungkasi, S. (2017). A simple but accurate explicit finite difference method for the advection-diffusion equation. Journal of Physics, 909, 012038–012038. https://doi.org/10.1088/1742-6596/909/1/012038

Sofiani, M. (2023). On parabolic partial differential equations with Hölder continuous diffusion coefficients. Journal of Mathematical Analysis and Applications, 526(1), 127224. https://doi.org/10.1016/j.jmaa.2023.127224

Weller, D. P., Bott, T. E., Caballero, M. D., & Irving, P. W. (2022). Development and illustration of a framework for computational thinking practices in introductory physics. Physical Review, 18(2). https://doi.org/10.1103/physrevphyseducres.18.020106

Yao, S., Gu, W., Wu, J., Lu, H., Zhang, S., Zhou, Y., & Lu, S. (2022). Dynamic energy flow analysis of the heat-electricity integrated energy systems with a novel decomposition-iteration algorithm. Applied Energy, 322, 119492. https://doi.org/10.1016/j.apenergy.2022.119492

Yong, W.-A. (2020). Intrinsic properties of conservation-dissipation formalism of irreversible thermodynamics. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 378(2170), 20190177. https://doi.org/10.1098/rsta.2019.0177

Zhang, B., & Yang, C. (2024). Discovering physics-informed neural networks model for solving partial differential equations through evolutionary computation. Swarm and Evolutionary Computation, 88, 101589–101589. https://doi.org/10.1016/j.swevo.2024.101589

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Published

2024-04-10

How to Cite

Dwi Satrio, G., Dewi, N. P., Fadilah, R. N., & Palupi, Z. D. (2024). Enhancing Student Understanding of Heat Distribution in Metal Rods Through Interactive Learning Using Finite Difference Methods. Current STEAM and Education Research, 2(1), 19-34. https://doi.org/10.58797/cser.020102

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