An improved error inhibiting block one-step method with radial basis function approximation for ODEs

Abstract

An explicit error inhibiting block one-step method was developed by Ditkowski and Gottlieb in 2017. It is motivated by a class of one-step methods described in Shampine and Watts (Math Comp 23:731–740, 1969), which takes initial s step values to generate the next s step values at each step. The error inhibiting method shares the form of the diagonally implicit multistage integration method of type 3 introduced in Butcher (Appl Numer Math 11:347–363, 1993). With the explicit error inhibiting scheme, the convergence of the global error is one order higher than that of the local truncation error, while in general the global error decays with the same order as the local truncation error. In this work, we improve the explicit error inhibiting block one-step method in order to further enhance the convergence and accuracy of the method. The main idea is to adopt the radial basis functions as a reconstruction basis replacing the polynomial basis. The numerical results confirm that the proposed method is efficient in providing enhanced order of accuracy.