Adaptive Matérn RBF Adams–Bashforth–Moulton methods for initial value problems

We construct adaptive Matérn radial basis function (MA-RBF) modifications of the classical two-step Adams–Bashforth and one-step Adams–Moulton formulas for first-order initial value problems (IVPs). The proposed schemes are based on a C4-smooth Matérn kernel and recover the corresponding classical Adams formulas in the flat limit, when the shape parameter tends to zero. Taylor expansions of the local truncation errors yield explicit adaptive values, used in place of the squared shape parameter, that raise the order of both formulas from two to three. The resulting MA-RBF Adams–Bashforth formula and Adams–Bashforth–Moulton predictor–corrector scheme are tested on two nonlinear benchmark IVPs. The numerical results confirm the predicted third-order convergence and show accuracy gains over the corresponding classical Adams schemes.