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<p>The coupled Burgers’ equations have two important properties: nonlinearity and coupling. In this paper, nonlinear coupling terms are discretized into the three-term recurrence approximation formats by the two-dimensional Taylor formula. The time derivatives are discretized by the second-order backward difference method (BDF2), which produces a semi-discrete scheme for the coupled Burgers’ equations. Furthermore, the reproducing kernel interpolation collocation numerical scheme (RKICNS) is proposed for the approximate solution of the semi-discrete scheme in the spatial direction. The stability, convergence, and error estimates of the numerical solutions of the coupled Burgers’ equations are rigorously discussed. Numerical experiments are tested to verify the effectiveness, high precision and decrease of computational complexity of the method compared with other existing methods. Multiple parameters of the coupled Burgers’ equations are chosen, which includes the condition with the very large Reynolds numbers. It is worth mentioning that the proposed method performs well in simulating local extreme points.</p>

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A high-accuracy reproducing kernel approach for the coupled Burgers’ equations

  • 期刊论文

  • 发表日期：

    2026年2月3日

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  • Jiabao Yang,
  • Zongwei Li,
  • Jiaxuan Chi,
  • Huanmin Yao

摘要

The coupled Burgers’ equations have two important properties: nonlinearity and coupling. In this paper, nonlinear coupling terms are discretized into the three-term recurrence approximation formats by the two-dimensional Taylor formula. The time derivatives are discretized by the second-order backward difference method (BDF2), which produces a semi-discrete scheme for the coupled Burgers’ equations. Furthermore, the reproducing kernel interpolation collocation numerical scheme (RKICNS) is proposed for the approximate solution of the semi-discrete scheme in the spatial direction. The stability, convergence, and error estimates of the numerical solutions of the coupled Burgers’ equations are rigorously discussed. Numerical experiments are tested to verify the effectiveness, high precision and decrease of computational complexity of the method compared with other existing methods. Multiple parameters of the coupled Burgers’ equations are chosen, which includes the condition with the very large Reynolds numbers. It is worth mentioning that the proposed method performs well in simulating local extreme points.