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An economical robust algorithm for solving 1D coupled Burgers’ equations in a semi-Lagrangian framework

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  • Park, Sangbeom
  • Kim, Philsu
  • Jeon, Yonghyeon
  • Bak, Soyoon

Abstract

In this study, we propose an efficient algorithm for solving one-dimensional coupled viscous Burgers’ equations. One of the main accomplishments of this study is to develop a stable high-order algorithm for the system of reaction–diffusion equations. The algorithm is “robust” because it is designed to prevent non-physical oscillations through an iteration procedure of a block Gauss-Seidel type. The other is to develop an efficient algorithm for the Cauchy problem. For this, we first find half of the upstream points by adopting a multi-step qth-order (q=2,3) error correction method. The algorithm is also “economical” in the sense that an interpolation strategy for finding the remaining upstream points is designed to dramatically reduce the high computational cost for solving the nonlinear Cauchy problem without damage to the order of accuracy. Three benchmark problems are simulated to investigate the accuracy and the superiority of the proposed method. It turns out that the proposed method numerically has the qth-order temporal and 4th-order spatial accuracies. In addition, the numerical experiments show that the proposed method is superior to the compared methods in the sense of the computational cost.

Suggested Citation

  • Park, Sangbeom & Kim, Philsu & Jeon, Yonghyeon & Bak, Soyoon, 2022. "An economical robust algorithm for solving 1D coupled Burgers’ equations in a semi-Lagrangian framework," Applied Mathematics and Computation, Elsevier, vol. 428(C).
    • Handle: RePEc:eee:apmaco:v:428:y:2022:i:c:s0096300322002594
    DOI: 10.1016/j.amc.2022.127185
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    References listed on IDEAS

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    1. Li, Qianhuan & Chai, Zhenhua & Shi, Baochang, 2015. "A novel lattice Boltzmann model for the coupled viscous Burgers’ equations," Applied Mathematics and Computation, Elsevier, vol. 250(C), pages 948-957.
    2. Doğan Kaya, 2001. "An explicit solution of coupled viscous Burgers' equation by the decomposition method," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 27, pages 1-6, January.
    3. Mohanty, R.K. & Dai, Weizhong & Han, Fei, 2015. "Compact operator method of accuracy two in time and four in space for the numerical solution of coupled viscous Burgers’ equations," Applied Mathematics and Computation, Elsevier, vol. 256(C), pages 381-393.
    4. Chen, Aihua & Li, Xuemei, 2006. "Darboux transformation and soliton solutions for Boussinesq–Burgers equation," Chaos, Solitons & Fractals, Elsevier, vol. 27(1), pages 43-49.
    5. Lai, Huilin & Ma, Changfeng, 2014. "A new lattice Boltzmann model for solving the coupled viscous Burgers’ equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 395(C), pages 445-457.
    6. Kim, Philsu & Kim, Dojin, 2020. "Convergence and stability of a BSLM for advection-diffusion models with Dirichlet boundary conditions," Applied Mathematics and Computation, Elsevier, vol. 366(C).
    7. Soliman, A.A., 2006. "The modified extended tanh-function method for solving Burgers-type equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 361(2), pages 394-404.
    8. Chen, Changkai & Zhang, Xiaohua & Liu, Zhang, 2020. "A high-order compact finite difference scheme and precise integration method based on modified Hopf-Cole transformation for numerical simulation of n-dimensional Burgers’ system," Applied Mathematics and Computation, Elsevier, vol. 372(C).
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