A collocation method based on one‐dimensional RBF interpolation scheme for solving PDEs

Mai‐Duy, N.; Tanner, Roger I.

doi:10.1108/09615530710723948

Cited by 42 publications

(35 citation statements)

References 26 publications

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“…[8,9]). Recently, an approximation scheme, which is based on point collocation, Cartesian grids and one-dimensional integrated RBF networks (1D-IRBFNs), has been proposed in [10,11]. A problem domain, which can be regular or irregular, is discretised by a Cartesian grid.…”

Section: Introductionmentioning

confidence: 99%

An integrated-RBF technique based on Galerkin formulation for elliptic differential equations

Mai‐Duy

Tran-Cong

2009

Engineering Analysis with Boundary Elements

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This paper presents a new radial-basis-function (RBF) technique for solving elliptic differential equations (DEs). The RBF solutions are sought to satisfy (a) the boundary conditions in a local sense using the point-collocation formulation, and (b) the governing equation in a global sense using the Galerkin formulation. In contrast to Galerkin finite-element techniques, the present Neumann boundary conditions are imposed in an exact manner. Unlike conventional RBF techniques, the present RBF approximations are constructed "locally" on grid lines through integration and they are

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Section: Introductionmentioning

confidence: 99%

An integrated-RBF technique based on Galerkin formulation for elliptic differential equations

Mai‐Duy

Tran-Cong

2009

Engineering Analysis with Boundary Elements

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“…In this section, we briefly describe the 1D-IRBFN methods [11] including 1D-IRBFN-2 and 1D-IRBFN-4 schemes, with the full details given in Appendix B. The domain of interest is discretised using a Cartesian grid, i.e.…”

Section: Numerical Approach: One-dimensional Radial Basis Function Nementioning

confidence: 99%

“…Our main purpose in the present paper is to justify the averaged model, deduced by the centre manifolds, by direct comparison of numerical solutions of the averaged (1-D) and original (2-D) models, using the one-dimensional integrated radial basis function network (1D-IRBFN) method [11]. The 1D-IRBFN and IRBFN-based methods have been successfully developed and applied to several engineering problems such as structural analysis [12,13], viscous and viscoelastic flows [14,15,16], and fluid-structure interaction [17].…”

Section: Introductionmentioning

confidence: 99%

Modelling dispersion in laminar and turbulent flows in an open channel based on centre manifolds using 1D-IRBFN method

Mohammed

Ngo-Cong

Strunin

et al. 2014

Applied Mathematical Modelling

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Centre manifold method is an accurate approach for analytically constructing an advection-diffusion equation (and even more accurate equations involving higher-order derivatives) for the depth-averaged concentration of substances in channels. This paper presents a direct numerical verification of converge to each other, with their velocities becoming practically equal. The obtained numerical results also demonstrate that the longitudinal diffusion can be neglected compared to the advection.

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“…Meth. Fluids 5 tion method for the solution of second-and fourth-order PDEs was presented by Mai-Duy and Tanner [23]. In this method, Cartesian grids were used to discretise both rectangular and non-rectangular problem domains.…”

Section: Introductionmentioning

confidence: 99%

Local moving least square‐one‐dimensional integrated radial basis function networks technique for incompressible viscous flows

Ngo-Cong

Mai‐Duy

Karunasena

et al. 2012

Numerical Methods in Fluids

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This paper presents a local moving least square -one-dimensional integrated radial basis function networks (LMLS-1D-IRBFN) method for solving incompressible viscous flow problems using stream function-vorticity formulation. In this method, the partition of unity method is employed as a framework to incorporate the moving least square (MLS) and one-dimensional integrated radial basis function networks (1D-IRBFN) techniques. The major advantages of the proposed method include: (i) a banded sparse system matrix which helps reduce the computational cost; (ii) the Kronecker-δ property of the constructed shape function which helps impose the essential boundary condition in an exact manner; and (iii) high accuracy and fast convergence rate owing to the use of integration instead of conventional differentiation to construct the local RBF approximations. Several examples including two-dimensional Poisson problems, lid-driven cavity flow and flow past a circular cylinder are considered and the present results are compared with the exact solutions and numerical results from other methods in the literature to demonstrate the attractiveness of the proposed method.

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A collocation method based on one‐dimensional RBF interpolation scheme for solving PDEs

Cited by 42 publications

References 26 publications

An integrated-RBF technique based on Galerkin formulation for elliptic differential equations

An integrated-RBF technique based on Galerkin formulation for elliptic differential equations

Modelling dispersion in laminar and turbulent flows in an open channel based on centre manifolds using 1D-IRBFN method

Local moving least square‐one‐dimensional integrated radial basis function networks technique for incompressible viscous flows

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