A variable inverse-multiquadric shape parameter applied with a meshless method for nonlinear PDEs

Abstract

It is known that all RBF-based meshfree methods suffer from a lack of reliable judgment on the choice of shape parameter, appearing in most of the RBFs. Many attempts on providing promisingly useful information on how to define this parameter have been made during the part years. Nevertheless, it is known that it is practically impossible to have one form of parameter that yields reasonable results for all kinds of problems. In this work, we aim to complete three important tasks. Firstly, we apply the methodology of RBF-collocation method, one of the classical form of meshfree/meshless schemes to PDEs with nonlinear nature. Secondly, several forms of variable Multiquadric shape parameters proposed and those available in literature have been gathered and applied to the same type of PDEs. Lastly, we propose a new form of shape parameter under the form of inverse-multiquadric type of RBF. In order to justify the quality of our proposed variable, all forms gathered from the literature are applied to the same PDEs, and all the results are compared against one another. The information gathered and presented in this work shall be useful for the future users in making decision and will provide useful guide to further invent another potential shape adaptive approaches as well.