Adaptive IQ and IMQ-RBFs for solving Initial Value Problems: Adam-Bashforth and Adam-Moulton methods
Samala Rathan, Deepit Shah, T. Hemanth Kumar, K. Sandeep Charan
TL;DR
This work addresses solving first-order IVPs by marrying adaptive inverse-quadratic and inverse multi-quadratic radial basis function interpolation with Adam-Bashforth and Adam-Moulton multistep methods. By locally selecting the RBF shape parameter $\epsilon_n$ to cancel the leading truncation error, the proposed IQ-RBF and IMQ-RBF schemes achieve equal or higher order of convergence than the classical AB/AM methods, while preserving consistency and stability. The authors derive AB2/AB3/AM2/AM3 variants, provide explicit optimal-$\epsilon$ formulas (and finite-difference surrogates) to realize higher-order accuracy, establish stability regions, and validate the improvements with multiple IVP test problems. The results indicate substantial gains in accuracy and convergence rate, with a framework suitable for extension to nonuniform meshes and broader shape-parameter choices in future work.
Abstract
In this paper, our objective is primarily to use adaptive inverse-quadratic (IQ) and inverse-multi-quadratic (IMQ) radial basis function (RBF) interpolation techniques to develop an enhanced Adam-Bashforth and Adam-Moulton methods. By utilizing a free parameter involved in the radial basis function, the local convergence of the numerical solution is enhanced by making the local truncation error vanish. Consistency and stability analysis is presented along with some numerical results to back up our assertions. The accuracy and rate of convergence of each proposed technique are equal to or better than the original Adam-Bashforth and Adam-Moulton methods by eliminating the local truncation error thus, the proposed adaptive methods are optimal. We conclude that both IQ and IMQ-RBF methods yield an improved order of convergence than classical methods, while the superiority of one method depends on the method and the problem considered.