A High-order Compact Local Integrated-RBF Scheme for Steady-state Incompressible Viscous Flows in the Primitive Variables
N. Thai-Quang;  K. Le-Cao;  N. Mai-Duy;  T. Tran-Cong

doi:10.3970/cmes.2012.084.528
Source CMES: Computer Modeling in Engineering & Sciences, Vol. 84, No. 6, pp. 528-558, 2012
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Keywords viscous flow, primitive variables, Cartesian grid, integrated radial basis function, high-order approximation, compact local stencil.
Abstract This study is concerned with the development of integrated radial-basis-function (IRBF) method for the simulation of two-dimensional steady-state incompressible viscous flows governed by the pressure-velocity formulation on Cartesian grids. Instead of using low-order polynomial interpolants, a high-order compact local IRBF scheme is employed to represent the convection and diffusion terms. Furthermore, an effective boundary treatment for the pressure variable, where Neumann boundary conditions are transformed into Dirichlet ones, is proposed. This transformation is based on global 1D-IRBF approximators using values of the pressure at interior nodes along a grid line and first-order derivative values of the pressure at the two extreme nodes of that grid line. The performance of the proposed scheme is investigated numerically through the solution of several linear (analytic tests including Stokes flows) and non-linear (recirculating cavity flow driven by combined shear & body forces and lid-driven cavity flow) problems. Unlike the global 1D-IRBF scheme, the proposed method leads to a sparse system matrix. Numerical results indicate that (i) the present solutions are more accurate and converge faster with grid refinement in comparison with standard finite-difference results; and (ii) the proposed boundary treatment for the pressure is more effective than conventional direct application of the Neumann boundary condition.
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