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Direct Collocation with Reproducing Kernel Approximation for Two-Phase Coupling System in a Porous Enclosure

Author

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  • Judy P. Yang

    (Department of Civil Engineering, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan
    Department of Civil Engineering, National Chiao Tung University, Hsinchu 30010, Taiwan)

  • Yi-Shan Liao

    (Department of Civil Engineering, National Yang Ming Chiao Tung University, Hsinchu 30010, Taiwan)

Abstract

The direct strong-form collocation method with reproducing kernel approximation is introduced to efficiently and effectively solve the natural convection problem within a parallelogrammic enclosure. As the convection behavior in the fluid-saturated porous media involves phase coupling, the resulting system is highly nonlinear in nature. To this end, the local approximation is adopted in conjunction with Newton–Raphson method. Nevertheless, to unveil the performance of the method in the nonlinear analysis, only single thermal natural convection is of major concern herein. A unit square is designated as the model problem to investigate the parameters in the system related to the convergence; several benchmark problems are used to verify the accuracy of the approximation, in which the stability of the method is demonstrated by considering various initial conditions, disturbance of discretization, inclination, aspect ratio, and reproducing kernel support size. It is shown that a larger support size can be flexible in approximating highly irregular discretized problems. The derivation of explicit operators with two-phase variables in solving the nonlinear system using the direct collocation is carried out in detail.

Suggested Citation

  • Judy P. Yang & Yi-Shan Liao, 2021. "Direct Collocation with Reproducing Kernel Approximation for Two-Phase Coupling System in a Porous Enclosure," Mathematics, MDPI, vol. 9(8), pages 1-25, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:8:p:897-:d:538253
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    References listed on IDEAS

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    1. Berna Bülbül & Mehmet Sezer, 2013. "A New Approach to Numerical Solution of Nonlinear Klein-Gordon Equation," Mathematical Problems in Engineering, Hindawi, vol. 2013, pages 1-7, June.
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