A high-order convergence analysis for semi-Lagrangian scheme of the Burgers' equation

Philsu Kim; Seongook Heo; Dojin Kim; Philsu Kim; Seongook Heo; Dojin Kim

doi:10.3934/math.2023571

AIMS Mathematics

2023, Volume 8, Issue 5: 11270-11296. doi: 10.3934/math.2023571

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A high-order convergence analysis for semi-Lagrangian scheme of the Burgers' equation

1.
Department of Mathematics, Kyungpook National University, Daegu 41566, Korea
2.
Department of Mathematics, Dongguk University, Seoul 04620, Korea

Received: 03 January 2023 Revised: 23 February 2023 Accepted: 01 March 2023 Published: 13 March 2023
MSC : 65M06, 65M12

In this article, we provide a comprehensive convergence and stability analysis of a semi-Lagrangian scheme for solving nonlinear Burgers' equations with a high-order spatial discretization. The analysis is for the iteration-free semi-Lagrangian scheme comprising the second-order backward finite difference formula (BDF2) for total derivative and the fourth-order central finite difference for diffusion term along the trajectory. The main highlight of the study is to thoroughly analyze the order of convergence of the discrete $ \ell_2 $-norm error $ \mathcal{O}(h^2+\triangle x^4+ \triangle x^{p+1}/h) $ by managing the relationship between the local truncation errors from each discretization procedure and the interpolation properties with a symmetric high-order discretization of the diffusion term. Furthermore, stability is established by the uniform boundedness of the numerical solution using the discrete Grönwall's Lemma. We provide numerical examples to support the validity of the theoretical convergence and stability analysis for the propounded backward semi-Lagrangian scheme.
- backward semi-Lagrangian scheme,
- convergence analysis,
- BDF2,
- Burgers' equations
Citation: Philsu Kim, Seongook Heo, Dojin Kim. A high-order convergence analysis for semi-Lagrangian scheme of the Burgers' equation[J]. AIMS Mathematics, 2023, 8(5): 11270-11296. doi: 10.3934/math.2023571

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Abstract

In this article, we provide a comprehensive convergence and stability analysis of a semi-Lagrangian scheme for solving nonlinear Burgers' equations with a high-order spatial discretization. The analysis is for the iteration-free semi-Lagrangian scheme comprising the second-order backward finite difference formula (BDF2) for total derivative and the fourth-order central finite difference for diffusion term along the trajectory. The main highlight of the study is to thoroughly analyze the order of convergence of the discrete $ \ell_2 $-norm error $ \mathcal{O}(h^2+\triangle x^4+ \triangle x^{p+1}/h) $ by managing the relationship between the local truncation errors from each discretization procedure and the interpolation properties with a symmetric high-order discretization of the diffusion term. Furthermore, stability is established by the uniform boundedness of the numerical solution using the discrete Grönwall's Lemma. We provide numerical examples to support the validity of the theoretical convergence and stability analysis for the propounded backward semi-Lagrangian scheme.

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