Improved localized radial basis function collocation method for multi-dimensional convection-dominated problems

Yun, Dongfang; Hon, Y.C.

doi:10.1016/j.enganabound.2016.03.003

Cited by 32 publications

(8 citation statements)

References 49 publications

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“…In the context of the RBF-FD method it is desirable to have a common discretization scheme defined on a common node distribution for the entire set of variables, e.g., u, p and T , while upwind techniques, although employed [62,63,64], tend to be avoided in favour of other techniques [65], expecially when dealing with high-order, accurate RBF-FD discretizations.…”

Section: Stabilizationmentioning

confidence: 99%

Analysis of geometric uncertainties in CFD problems solved by RBF-FD meshless method

Zamolo

Parussini

2020

Journal of Computational Physics

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Section: Stabilizationmentioning

confidence: 99%

Analysis of geometric uncertainties in CFD problems solved by RBF-FD meshless method

Zamolo

Parussini

2020

Journal of Computational Physics

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“…If the values of any variable are known at the scattered points, the interpolation can then be used to calculate values at in-between locations within the cloud. However, if the values are unknown but satisfy an underlying differential equation, the governing equation can be satisfied discretely at the scattered points either by collocation or by the method of weighted residuals [50][51][52][53][54]. The resulting set of linear (or nonlinear) equations is then solved for the unknown values of the dependent variable at the scattered points.…”

Section: Introductionmentioning

confidence: 99%

A Non-Nested Multilevel Method for Meshless Solution of the Poisson Equation in Heat Transfer and Fluid Flow

Radhakrishnan,

Xu,

Shahane

et al. 2021

Preprint

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We present a non-nested multilevel algorithm for solving the Poisson equation discretized at scattered points using polyharmonic radial basis function (PHS-RBF) interpolations. We append polynomials to the radial basis functions to achieve exponential convergence of discretization errors. The interpolations are performed over local clouds of points and the Poisson equation is collocated at each of the scattered points, resulting in a sparse set of discrete equations for the unkown variables. To solve this set of equations, we have developed a non-nested multilevel algorithm utilizing multiple independently generated coarse sets of points. The restriction and prolongation operators are also constructed with the same RBF interpolations procedure.The performance of the algorithm for Dirichlet and all-Neumann boundary conditions is evaluated in three model geometries using a manufactured solution. For Dirichlet boundary conditions, rapid convergence is observed

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“…A large number of papers have presented numerical schemes for solving several types of partial differential equations (PDEs) utilizing LRBFs. Such PDEs appear, for example, in diffusion-convection problems with phase change [44], natural convection problems [26], natural convection under the influence of a static magnetic field [37], multi-dimensional convection-dominated problems [47], the system of second-order boundary value problems [16], polarized radiative transfer in participating media [42], thermoelasticity in two dimensions [35], convection-dominated fluid flow problems [47] and conservation laws equations [15].…”

Section: Introductionmentioning

confidence: 99%

A Meshless Local Galerkin Integral Equation Method for Solving a Type of Darboux Problems Based on Radial Basis Functions

Assari¹,

Asadi‐Mehregan

Dehghan

2021

ANZIAM J.

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The main goal of this paper is to solve a class of Darboux problems by converting them into the two-dimensional nonlinear Volterra integral equation of the second kind. The scheme approximates the solution of these integral equations using the discrete Galerkin method together with local radial basis functions, which use a small set of data instead of all points in the solution domain. We also employ the Gauss–Legendre integration rule on the influence domains of shape functions to compute the local integrals appearing in the method. Since the scheme is constructed on a set of scattered points and does not require any background meshes, it is meshless. The error bound and the convergence rate of the presented method are provided. Some illustrative examples are included to show the validity and efficiency of the new technique. Furthermore, the results obtained demonstrate that this method uses much less computer memory than the method established using global radial basis functions.

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Improved localized radial basis function collocation method for multi-dimensional convection-dominated problems

Cited by 32 publications

References 49 publications

Analysis of geometric uncertainties in CFD problems solved by RBF-FD meshless method

Analysis of geometric uncertainties in CFD problems solved by RBF-FD meshless method

A Non-Nested Multilevel Method for Meshless Solution of the Poisson Equation in Heat Transfer and Fluid Flow

A Meshless Local Galerkin Integral Equation Method for Solving a Type of Darboux Problems Based on Radial Basis Functions

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