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Numerical solutions of the coupled unsteady nonlinear convection-diffusion equations based on generalized finite difference method

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Abstract.

In this paper, the meshless Generalized Finite Difference Method (GFDM) in conjunction with the second-order explicit Runge-Kutta method (RK2 method) is presented to solve coupled unsteady nonlinear convection-diffusion equations (CDEs). Compared with the conventional Euler method, the RK2 method not only has higher accuracy but also reduces the possibility of numerical oscillation in time discretization, especially for the nonlinear and coupled cases. The generalized finite difference method, which is a localized collocation method, is famous for its simplicity and adaptability in the numerical solution of partial differential equations. Benefiting from Taylor series and moving least squares, its partial derivatives can be formed by a series of surrounding space points. In comparison with traditional finite difference methods, the proposed GFDM is free of mesh and available for irregular discretization nodes. In this study, the stencil selection algorithms are introduced to choose the stencil support of a certain node from the whole discretization nodes. Error analysis and numerical investigations are presented to demonstrate the effectiveness of the proposed GFDM for solving the coupled linear and nonlinear unsteady convection-diffusion equations. Then it is successfully applied to three benchmark examples of the coupled unsteady nonlinear CDEs encountered in the double-diffusive natural convection process, chemotaxis-haptotaxis model of cancer invasion, and thermo-hygro coupling model of concrete.

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Authors and Affiliations

  1. Key Laboratory of Coastal Disaster and Defence of Ministry of Education, Hohai University, 210098, Nan**g, China

    Zhuo-Jia Fu & Zhuo-Chao Tang

  2. Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, 211100, Nan**g, China

    Zhuo-Jia Fu, Zhuo-Chao Tang, Po-Wei Li & Timon Rabczuk

  3. State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, 116024, Dalian, China

    Zhuo-Jia Fu

  4. College of Civil Engineering, Hohai University, 210098, Nan**g, China

    Hai-Tao Zhao

  5. Institute of Structural Mechanics, Bauhaus-University Weimar, 99423, Weimar, Germany

    Timon Rabczuk

Authors
  1. Zhuo-Jia Fu
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  2. Zhuo-Chao Tang
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  3. Hai-Tao Zhao
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  4. Po-Wei Li
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  5. Timon Rabczuk
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Correspondence to Zhuo-Jia Fu.

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Fu, ZJ., Tang, ZC., Zhao, HT. et al. Numerical solutions of the coupled unsteady nonlinear convection-diffusion equations based on generalized finite difference method. Eur. Phys. J. Plus 134, 272 (2019). https://doi.org/10.1140/epjp/i2019-12786-7

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