A new high-order compact numerical method for the two-dimensional nonlinear Burgers' equation

Shengdi Wang; Jiaye Gan; Yingnan Qi; Lili Wu; Shengdi Wang; Jiaye Gan; Yingnan Qi; Lili Wu

doi:10.3934/era.2026063

Electronic Research Archive

2026, Volume 34, Issue 3: 1386-1412. doi: 10.3934/era.2026063

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Research article

A new high-order compact numerical method for the two-dimensional nonlinear Burgers' equation

1.
School of Mathematics and Computer Science, Ningxia Normal University, Guyuan 756000, China
2.
School of Power and Mechanical Engineering, Wuhan University, Wuhan 430072, China

Received: 04 November 2025 Revised: 23 January 2026 Accepted: 27 January 2026 Published: 11 February 2026

The Burgers' equation is a key model equation in fluid mechanics and nonlinear physics, widely used to describe the formation and propagation of shock waves. This paper proposes a highly accurate numerical algorithm to solve the two-dimensional Burgers' equation. In the algorithmic framework, the time derivative is discretized using a fourth-order backward finite difference approach. In contrast, the spatial derivatives are processed by a fourth-order Padé scheme and a fourth-order compact finite difference scheme, respectively. To linearize the nonlinear term in the equation, an extrapolation-based linearization technique is introduced, which effectively reduces the computational complexity while maintaining the numerical accuracy. Theoretical analyses of this algorithm demonstrated its unique solvability and stability. Numerous numerical experiments verify the scheme's stability, computational efficiency, and convergence. This approach achieves a fourth-order accuracy in both the temporal and spatial dimensions, thus offering the benefit of maintaining high agreement between the numerical and exact solutions, even when using relatively large time steps, thereby significantly improving the computational efficiency. In addition, the algorithm employs linearization to ease the computational demands, thereby providing a viable and efficient scheme to address high-dimensional nonlinear Burgers' equations.
Citation: Shengdi Wang, Jiaye Gan, Yingnan Qi, Lili Wu. A new high-order compact numerical method for the two-dimensional nonlinear Burgers' equation[J]. Electronic Research Archive, 2026, 34(3): 1386-1412. doi: 10.3934/era.2026063

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Abstract

The Burgers' equation is a key model equation in fluid mechanics and nonlinear physics, widely used to describe the formation and propagation of shock waves. This paper proposes a highly accurate numerical algorithm to solve the two-dimensional Burgers' equation. In the algorithmic framework, the time derivative is discretized using a fourth-order backward finite difference approach. In contrast, the spatial derivatives are processed by a fourth-order Padé scheme and a fourth-order compact finite difference scheme, respectively. To linearize the nonlinear term in the equation, an extrapolation-based linearization technique is introduced, which effectively reduces the computational complexity while maintaining the numerical accuracy. Theoretical analyses of this algorithm demonstrated its unique solvability and stability. Numerous numerical experiments verify the scheme's stability, computational efficiency, and convergence. This approach achieves a fourth-order accuracy in both the temporal and spatial dimensions, thus offering the benefit of maintaining high agreement between the numerical and exact solutions, even when using relatively large time steps, thereby significantly improving the computational efficiency. In addition, the algorithm employs linearization to ease the computational demands, thereby providing a viable and efficient scheme to address high-dimensional nonlinear Burgers' equations.

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