A fast and stabilized meshless method for the convection-dominated convection-diffusion problems

References

• E. Ortega, E. Oñate, S. Idelsohn, and R. Flores, Comparative Accuracy and Performance Assessment of the Finite Point Method in Compressible Flow Problems, Comput. Fluids. vol. 89, pp. 53–65, 2014.
   Web of Science ®Google Scholar
• M. N. Oliaei, A. Pak, and K. Soga, A Coupled Hydro-Mechanical Analysis for Prediction of Hydraulic Fracture Propagation in Saturated Porous Media Using EFG Mesh-Less Method, Comput. Geotech., vol. 55, pp. 254–266, 2014.
   Web of Science ®Google Scholar
• M. Dehghan and A. Ghesmati, Numerical Simulation of Two-Dimensional Sine-Gordon Solitons Via a Local Weak Meshless Technique Based on the Radial Point Interpolation Method (RPIM), Comput. Phy. Commun., vol. 181, pp. 772–786, 2010.
   Web of Science ®Google Scholar
• A. Abdollahifar, M. R. Nami, and A. R. Shafiei, A New MLPG Method for Elastostatic Problems, Eng. Anal. Bound. Elem., vol. 36, pp. 451–457, 2012.
   Web of Science ®Google Scholar
• E. H. Ooi and V. Popov, Meshless Solution of Axisymmetric Convection-Diffusion Equation: A Comparison Between Two Alternative RBIE Implementations, Eng. Anal. Bound. Elem., vol. 37, pp. 719–727, 2013.
   Web of Science ®Google Scholar
• H. Lin and S. N. Atluri, Meshless Local Petrov-Galerkin (MLPG) Method for Convection-Diffusion Problems, Comput. Model. Eng. Sci., vol. 1, pp. 45–60, 2000.
   Web of Science ®Google Scholar
• X. H. Wu, Z. J. Chang, Y. L. Lu, W. Q. Tao, and S. P. Shen, An Analysis of the Convection–Diffusion Problems Using Meshless and Meshbased Methods, Eng. Anal. Bound. Elem., vol. 36, pp. 1040–1048, 2012.
   Web of Science ®Google Scholar
• X. H. Zhang, J. Ouyang, and J. Y. Wang, Stabilization Meshless Method for Convection Dominated Problems, Appl. Math. Mech., vol. 29, pp. 1067–1075, 2008.
   Web of Science ®Google Scholar
• Y. T. Gu and G. R. Liu, Meshless Techniques for Convection Dominated Problems, Comput. Mech., vol. 38, pp. 171–182, 2006.
   Web of Science ®Google Scholar
• X. Q. Qin, X. B. Duan, and W. Guo, The Characteristic Radial Basis Meshless Method for Convection-Dominated Diffusion Equations, Appl. Mech. Mater., vol. 182–183, pp. 1756–1760, 2012.
   Google Scholar
• C. Varanasi, J. Y. Murthy, and S. Mathur, Numerical Schemes for the Convection-Diffusion Equation Using a Meshless Finite-Difference Method, Numer. Heat Transfer, Part B., vol. 62, pp. 1–27, 2012.
   Web of Science ®Google Scholar
• S. Marras, J. F. Kelly, F. X. Giraldo, and M. Vázquez, Variational Multiscale Stabilization of High-Order Spectral Elements for the Advection–Diffusion Equation, J. Comput. Phy., vol. 231, pp. 7187–7213, 2012.
   Web of Science ®Google Scholar
• L. Zhang, J. Ouyang, X. X. Wang, and X. H. Zhang, Variational Multiscale Element-Free Galerkin Method for 2D Burgers’ Equation, J. Comput. Phys., vol. 229, pp. 7147–7161, 2010.
   Web of Science ®Google Scholar
• L. Zhang, J. Ouyang, X. H. Zhang, and W. B. Zhang,. On a Multi-Scale Element-Free Galerkin Method for the Stokes Problem, Appl. Math. Comput., vol. 203, pp. 745–753, 2008.
   Web of Science ®Google Scholar
• L. Zhang, J. Ouyang, T. Jiang, and C. L. Ruan, Variational Multiscale Element Free Galerkin Method for the Water Wave Problems, J. Comput. Phys., vol. 230, pp. 5045–5060, 2011.
   Web of Science ®Google Scholar
• L. Zhang, J. Ouyang, and X. H. Zhang, The Variational Multiscale Element Free Galerkin Method for MHD Flows at High Hartmann Numbers, Comput. Phys. Commun., vol. 184, pp. 1106–1118, 2013.
   Web of Science ®Google Scholar
• X. H. Zhang and H. Xiang, Variational Multiscale Element Free Galerkin Method for Convection-Diffusion-Reaction Equation with Small Diffusion, Eng. Anal. Bound. Elem., vol. 46, pp. 85–92, 2014.
   Web of Science ®Google Scholar
• X. H. Zhang and P. Zhang, Meshless Modeling of Natural Convection Problems in Non-Rectangular Cavity Using the Variational Multiscale Element Free Galerkin Method, Eng. Anal. Bound. Elem., vol. 61, pp. 287–300, 2015.
   Web of Science ®Google Scholar
• K. Pearson, On Lines and Planes of Closest Fit to Systems of Points In Space, Mag. J. Sci., vol. 2, pp. 559–572, 1901.
   Google Scholar
• M. F. Barone, I. Kalashnikova, D. J. Segalman, and H. K. Thornquist, Stable Galerkin Reduced Order Models for Linearized Compressible Flow, J. Comput. Phy., vol. 228, pp. 1932–1946, 2009.
   Web of Science ®Google Scholar
• S. Walton, O. Hassan, and K. Morgan, Reduced Order Modelling for Unsteady Fluid Flow Using Properorthogonal Decomposition and Radial Basis Functions, Appl. Math. Modell., vol. 37, pp. 8930–8945, 2013.
   Web of Science ®Google Scholar
• D. Binion, X. L. Chen, and A. Krylov, Enhanced Proper Orthogonal Decomposition Method for Efficient Nonlinear Model Reduction, Finite. Elem. Anal. Des., vol. 47, pp. 728–738, 2011.
   Web of Science ®Google Scholar
• T. A. Brenner, R. L. Fontenot, P. G. A. Cizmas, T. J. O'Brien, and R. W. Breault, A Reduced-Order Model for Heat Transfer in Multiphase Flow and Practical Aspects of the Proper Orthogonal Decomposition, Comput. Chem. Eng., vol. 43, pp. 68–80, 2012.
   Web of Science ®Google Scholar
• A. M. Rehm, E. Y. Scribner, and H. M. Fathallah-Shaykh, Proper Orthogonal Decomposition for Parameter Estimation in Oscillating Biological Networks, J. Comput. Appl. Math., vol. 258, pp. 135–150, 2014.
   Web of Science ®Google Scholar
• X. L. Bill and X. Hu, Proper Orthogonal Decomposition Reduced Model for Mass Transport in Heterogenous Media, Stoch Environ Res Risk Assess., vol. 27, pp. 1181–1191, 2013.
   Web of Science ®Google Scholar
• B. A. Freno, T. A. Brenner, and P. G. A. Cizmas, Using Proper Orthogonal Decomposition to Model Off-Reference Flow Conditions, Int. J. Non-Linear Mech. vol. 54, pp. 76–84, 2013.
   Web of Science ®Google Scholar
• V. Buljak, Proper Orthogonal Decomposition and Radial Basis Functions for Fast Simulations, in V. Buljak, (eds.), Inverse Analysis with Model Reduction, chap. 3, Springer-Verlag, Berlin Heidelberg, 2012.
   Google Scholar
• G. Kerschen, J. Golinval, A. F. Vakakis, and L. A. Bergman, The Method of Proper Orthogonal Decomposition for Dynamical Characterization and Order Reduction of Mechanical Systems: An Overview, Nonlinear Dynam., vol. 41, pp. 147–169, 2005.
   Web of Science ®Google Scholar
• X. H. Zhang and H. Xiang, A Fast Meshless Method Based on Proper Orthogonal Decomposition for the Transient Heat Conduction Problems, Int. J. Heat Mass Transfer., vol. 84, pp. 729–739, 2015.
   Web of Science ®Google Scholar
• H. Xiang and X. H. Zhang, Variational Multiscale Element-Free Galerkin Method and Precise Time Step Integration Method for Convection-Diffusion Problems, Numer. Heat Transfer, Part A. vol. 67, pp. 210–223, 2015.
   Web of Science ®Google Scholar