Multistep scattered data interpolation using compactly supported radial basis functions
Journal of Computational and Applied Mathematics - Special issue on scattered data
An efficient numerical scheme for Burgers' equation
Applied Mathematics and Computation
Compactly supported radial basis functions for shallow water equations
Applied Mathematics and Computation
Overlapping domain decomposition method by radial basis functions
Applied Numerical Mathematics
Scattered node compact finite difference-type formulas generated from radial basis functions
Journal of Computational Physics
A Stable Algorithm for Flat Radial Basis Functions on a Sphere
SIAM Journal on Scientific Computing
Computers & Mathematics with Applications
Meshfree Approximation Methods with MATLAB
Meshfree Approximation Methods with MATLAB
The use of PDE centres in the local RBF Hermitian method for 3D convective-diffusion problems
Journal of Computational Physics
Stable computation of multiquadric interpolants for all values of the shape parameter
Computers & Mathematics with Applications
Meshfree explicit local radial basis function collocation method for diffusion problems
Computers & Mathematics with Applications
The role of the multiquadric shape parameters in solving elliptic partial differential equations
Computers & Mathematics with Applications
Optimal variable shape parameter for multiquadric based RBF-FD method
Journal of Computational Physics
Journal of Computational Physics
Multivariate interpolation with increasingly flat radial basis functions of finite smoothness
Advances in Computational Mathematics
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This work proposes an alternative decomposition for local scalable meshless RBF collocation. The proposed method operates on a dataset of scattered nodes that are placed within the solution domain and on the solution boundary, forming a small RBF collocation system around each internal node. Unlike other meshless local RBF formulations that are based on a generalised finite difference (RBF-FD) principle, in the proposed ''finite collocation'' method the solution of the PDE is driven entirely by collocation of PDE governing and boundary operators within the local systems. A sparse global collocation system is obtained not by enforcing the PDE governing operator, but by assembling the value of the field variable in terms of the field value at neighbouring nodes. In analogy to full-domain RBF collocation systems, communication between stencils occurs only over the stencil periphery, allowing the PDE governing operator to be collocated in an uninterrupted manner within the stencil interior. The local collocation of the PDE governing operator allows the method to operate on centred stencils in the presence of strong convective fields; the reconstruction weights assigned to nodes in the stencils being automatically adjusted to represent the flow of information as dictated by the problem physics. This ''implicit upwinding'' effect mitigates the need for ad-hoc upwinding stencils in convective dominant problems. Boundary conditions are also enforced within the local collocation systems, allowing arbitrary boundary operators to be imposed naturally within the solution construction. The performance of the method is assessed using a large number of numerical examples with two steady PDEs; the convection-diffusion equation, and the Lame-Navier equations for linear elasticity. The method exhibits high-order convergence in each case tested (greater than sixth order), and the use of centred stencils is demonstrated for convective-dominant problems. In the case of linear elasticity, the stress fields are reproduced to the same degree of accuracy as the displacement field, and exhibit the same order of convergence. The method is also highly stable towards variations in basis function flatness, demonstrating significantly improved stability in comparison to finite-difference type RBF collocation methods.