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Proceedings of the 4th International Conference on Computing Innovation and Applied Physics, CONF-CIAP 2025, 17-23 January 2025, Eskişehir, Turkey

Research Article

Numerical Schemes for Partial Difference Equation in Physics

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  • @INPROCEEDINGS{10.4108/eai.17-1-2025.2355239,
        author={Houxu  Chen and Shengjie  Niu and Shuming  Zhang},
        title={Numerical Schemes for Partial Difference Equation in  Physics},
        proceedings={Proceedings of the 4th International Conference on Computing Innovation and Applied Physics, CONF-CIAP 2025, 17-23 January 2025, Eskişehir, Turkey},
        publisher={EAI},
        proceedings_a={CONF-CIAP},
        year={2025},
        month={4},
        keywords={numerical scheme diffusion equation advection equation partial differential  equation},
        doi={10.4108/eai.17-1-2025.2355239}
    }
    
  • Houxu Chen
    Shengjie Niu
    Shuming Zhang
    Year: 2025
    Numerical Schemes for Partial Difference Equation in Physics
    CONF-CIAP
    EAI
    DOI: 10.4108/eai.17-1-2025.2355239
Houxu Chen1,*, Shengjie Niu2, Shuming Zhang3
  • 1: Fudan University, Shanghai, China
  • 2: University of Edinburgh, UK
  • 3: University College London, UK
*Contact email: 21307130276@m.fudan.edu.cn

Abstract

This paper examines various numerical schemes for 1-D and 2-D advection and diffusion equations using MATLAB, focusing on stability, accuracy, and performance under different boundary conditions [1]. For 1-D advection, methods such as the upwind, implicit up wind, Beam-Warming(B-W), Lax-Friedrichs(L-F), and Lax-Wendroff(L-W) schemes are evaluated. The implicit upwind scheme delivers consistent results, the Beam-Warming scheme works well under specific conditions, while the upwind scheme shows dissipation and dispersion. In 2-D advection, the upwind and Lax-Friedrichs schemes are tested, with the upwind scheme being more stable with discontinuities but less stable for smooth solutions. For diffusion, the Classical, Dufort-Frankel(D-F), and Crank-Nicolson(C-N) schemes are analyzed. The Crank-Nicolson scheme proves to be the most accurate, while the Classical scheme is fast but mesh-dependent, and the Dufort-Frankel scheme is stable but introduces minor fluctuations. The paper suggests using operator splitting to improve 2-D advection stability.

Keywords
numerical scheme diffusion equation advection equation partial differential equation
Published
2025-04-07
Publisher
EAI
http://dx.doi.org/10.4108/eai.17-1-2025.2355239
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