Adaptive radial basis function (RBF) methods have been developed recently in Gu and Jung (2020) based on the multiquadric (MQ) RBFs for solving initial value problems (IVPs). The proposed adaptive RBF methods utilize the free parameter in order to adaptively enhance the local convergence of the numerical solution. Methods pertaining to the polynomial interpolation yield only fixed rate of convergence regardless of the solution smoothness while the proposed methods use the smoothness of the solution, given in derivatives of the solution, to control the rate of convergence. In this paper, for the completion of the development of the adaptive RBF methods, we develop various adaptive Gaussian RBF methods for solving IVPs by modifying the classical solvers such as the Euler’s method, midpoint method, Adams–Bashforth method and Adams–Moulton method by replacing the polynomial basis with the Gaussian RBFs. For each development, we compare the performance with the adaptive MQ-RBF methods and explain when and why the adaptive Gaussian methods are better or not than the MQ-RBF ones. We provide the collection of modifications with the MQ and Gaussian RBFs. We also provide the stability regions for the adaptive Gaussian methods. Numerical results confirm that the adaptivity enhances accuracy and convergence and also show the differences and similarities between MQ and Gaussian RBFs in their performance — we found that the adaptive MQ-RBF method has larger stability region than the Gaussian RBF method. Both MQ and Gaussian RBF methods yield the desired order of convergence while the superiority of one method to the other depends on the method and the problem considered.

Gu和Jung（2020）最近基于多二次（MQ）RBF开发了自适应径向基函数（RBF）方法，用于解决初值问题（IVP）。所提出的自适应RBF方法利用自由参数，以自适应地增强数值解的局部收敛性。涉及多项式插值的方法仅产生固定的收敛速度，而与解决方案的平滑度无关，而所提出的方法使用在求解的导数中给出的解决方案的平滑度来控制收敛速度。在本文中，为了完成自适应RBF方法的开发，我们通过修改经典的求解器（例如Euler方法，中点方法，Adams–Bashforth方法和Adams–Moulton方法，用高斯RBF代替多项式基础。对于每个开发，我们将性能与自适应MQ-RBF方法进行比较，并说明何时以及为什么自适应高斯方法比MQ-RBF方法更好或更不好。我们提供了MQ和高斯RBF的修改集。我们还为自适应高斯方法提供了稳定区域。数值结果证实了自适应性提高了准确性和收敛性，并且还显示了MQ和高斯RBF在性能上的异同—我们发现，自适应MQ-RBF方法比高斯RBF方法具有更大的稳定性区域。