A compact five-point stencil based on integrated RBFs for 2D second-order differential problems

Authors:
N. Mai-Duy;T. Tran-Cong
Affiliations:
Computational Engineering and Science Research Centre, Faculty of Engineering and Surveying, University of Southern Queensland, Toowoomba, QLD 4350, Australia;Computational Engineering and Science Research Centre, Faculty of Engineering and Surveying, University of Southern Queensland, Toowoomba, QLD 4350, Australia
Venue:
Journal of Computational Physics
Year:
2013

Citing 8
Cited 0

Numerical solution of differential equations using multiquadric radial basis function networks

Neural Networks
Scattered node compact finite difference-type formulas generated from radial basis functions

Journal of Computational Physics
Local radial basis function based gridfree scheme for unsteady incompressible viscous flows

Journal of Computational Physics
Meshfree Approximation Methods with MATLAB

Meshfree Approximation Methods with MATLAB
Stable computation of multiquadric interpolants for all values of the shape parameter

Computers & Mathematics with Applications
Multiquadric collocation method with integralformulation for boundary layer problems

Computers & Mathematics with Applications
Compact local integrated-RBF approximations for second-order elliptic differential problems

Journal of Computational Physics
An implicit RBF meshless approach for time fractional diffusion equations

Computational Mechanics

Quantified Score

Hi-index	31.45

Visualization

Abstract

In this paper, a compact 5-point stencil for the discretisation of second-order partial differential equations (PDEs) in two space dimensions is proposed. We employ integrated radial basis functions in one dimension (1D-IRBFs) to construct the approximations for the dependent variable and its derivatives over the three nodes in each direction of the stencil. Certain nodal values of the second-order derivatives are incorporated into the approximations with the help of the integration constants. In the case of elliptic PDEs, one algebraic equation is formed at each interior node, and the obtained final system, of which each row has 5 non-zero entries, is solved iteratively using a Picard scheme. In the case of parabolic PDEs discretised with a Crank-Nicolson procedure, a set of three simultaneous algebraic equations is established at each interior node and the three equations are then combined to form two tridiagonal equations through the implicit elimination approach. Linear and non-linear test problems, including lid-driven cavity flow and natural convection between the outer square and the inner cylinder, are considered to verify the proposed stencil.