Numerical analysis and simulation of the compact difference scheme for the pseudo-parabolic Burgers' equation

Yunxia Niu; Chaoran Qi; Yao Zhang; Wahidullah Niazi; Yunxia Niu; Chaoran Qi; Yao Zhang; Wahidullah Niazi

doi:10.3934/era.2025080

Electronic Research Archive

2025, Volume 33, Issue 3: 1763-1791. doi: 10.3934/era.2025080

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Numerical analysis and simulation of the compact difference scheme for the pseudo-parabolic Burgers' equation

1.
Department of Mathematics, School of Science, Xinjiang Institute of Technology, Akesu 843100, China
2.
Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou 310018, China
3.
Zhejiang Province Institute of Architectural Design and Research, Hangzhou 310006, China
4.
Department of Mathematic, Samangan University, Samangan 2001, Afghanistan

Received: 21 January 2025 Revised: 14 March 2025 Accepted: 18 March 2025 Published: 26 March 2025

In this paper, we analyzed and tested a nonlinear implicit compact finite difference scheme for the pseudo-parabolic Burgers' equation. The discrete conservation laws and boundedness of the scheme were rigorously established. We then proved the unique solvability of the numerical scheme by reformulating it as an equivalent system. Furthermore, using the energy method, we derived an error estimate for the proposed scheme, achieving a convergence order of $ {\mathcal{O}}(\tau^2 + h^4) $ under the discrete $ L^\infty $-norm. The stability of the compact finite difference scheme was subsequently proven using a similar approach. Finally, a series of numerical experiments were performed to validate the theoretical findings.
- pseudo-parabolic Burgers' equation,
- compact difference scheme,
- pointwise error estimate,
- stability,
- $ L^{\infty} $-norm
Citation: Yunxia Niu, Chaoran Qi, Yao Zhang, Wahidullah Niazi. Numerical analysis and simulation of the compact difference scheme for the pseudo-parabolic Burgers' equation[J]. Electronic Research Archive, 2025, 33(3): 1763-1791. doi: 10.3934/era.2025080

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Abstract

In this paper, we analyzed and tested a nonlinear implicit compact finite difference scheme for the pseudo-parabolic Burgers' equation. The discrete conservation laws and boundedness of the scheme were rigorously established. We then proved the unique solvability of the numerical scheme by reformulating it as an equivalent system. Furthermore, using the energy method, we derived an error estimate for the proposed scheme, achieving a convergence order of $ {\mathcal{O}}(\tau^2 + h^4) $ under the discrete $ L^\infty $-norm. The stability of the compact finite difference scheme was subsequently proven using a similar approach. Finally, a series of numerical experiments were performed to validate the theoretical findings.

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