Stable integrated RBF calculation using preconditioning and high-order compact approximation for second-order PDEs

Stable integrated RBF calculation using preconditioning and high-order compact approximation for second-order PDEs

It is well known that the accuracy of several radial basis function (RBF) methods, including those based on multiquadric (MQ) RBFs, depends on a free shape parameter. For smooth solutions, it is theoretically claimed that without round-off error the highest accuracy for a given number of nodal points is regularly achieved with extreme values of the shape parameter, where the RBFs become increasingly flat. However, the limit values of the shape parameter (increasingly flat) often leads to very ill-conditioned problems. To alleviate this difficulty, we present a RBF method for solving second-order PDEs, in which (i) the RBF approximations are constructed using the integral approach where the starting points are fourth-order derivatives; and (ii) a simple but effective preconditioning technique is employed in the process of converting the RBF coefficient space into the physical space. In this paper, we first numerically study the effect of the shape parameter on the solution accuracy of the present RBF method through Poisson equation; and, then apply the method employed with extreme values of the shape parameter to simulate several fluid flow problems where highly accurate and stable solutions are produced.

Author assigns all right, title and interest throughout the world in and to the copyright in the print and electronic forms of the Work to the Publisher.

Numerical methods for solving PDEs using integrated radial basis function (author's note)