Pointwise error estimate of conservative difference scheme for supergeneralized viscous Burgers' equation

Yang Shi; Xuehua Yang; Yang Shi; Xuehua Yang

doi:10.3934/era.2024068

Electronic Research Archive

2024, Volume 32, Issue 3: 1471-1497. doi: 10.3934/era.2024068

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Pointwise error estimate of conservative difference scheme for supergeneralized viscous Burgers' equation

Yang Shi ,
Xuehua Yang ^,

School of Science, Hunan University of Technology, Zhuzhou 412007, China

Academic Editor: Michael Winkler

Received: 15 December 2023 Revised: 28 January 2024 Accepted: 31 January 2024 Published: 06 February 2024

This work focuses on exploring pointwise error estimate of three-level conservative difference scheme for supergeneralized viscous Burgers' equation. The cut-off function method plays an important role in constructing difference scheme and presenting numerical analysis. We study the conservative invariant of proposed method, which is energy-preserving for all positive integers $ p $ and $ q $. Meanwhile, one could apply the discrete energy argument to the rigorous proof that the three-level scheme has unique solution combining the mathematical induction. In addition, we prove the $ L_2 $-norm and $ L_{\infty} $-norm convergence of proposed scheme in pointwise sense with separate and different ways, which is different from previous work in ^[1]. Numerical results verify the theoretical conclusions.
- supergeneralized viscous Burgers' equation,
- finite difference method,
- conservativity,
- pointwise error estimate,
- convergence
Citation: Yang Shi, Xuehua Yang. Pointwise error estimate of conservative difference scheme for supergeneralized viscous Burgers' equation[J]. Electronic Research Archive, 2024, 32(3): 1471-1497. doi: 10.3934/era.2024068

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Abstract

This work focuses on exploring pointwise error estimate of three-level conservative difference scheme for supergeneralized viscous Burgers' equation. The cut-off function method plays an important role in constructing difference scheme and presenting numerical analysis. We study the conservative invariant of proposed method, which is energy-preserving for all positive integers $ p $ and $ q $. Meanwhile, one could apply the discrete energy argument to the rigorous proof that the three-level scheme has unique solution combining the mathematical induction. In addition, we prove the $ L_2 $-norm and $ L_{\infty} $-norm convergence of proposed scheme in pointwise sense with separate and different ways, which is different from previous work in ^[1]. Numerical results verify the theoretical conclusions.

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