Adaptive Gaussian radial basis function methods for initial value problems: Construction and comparison with adaptive multiquadric radial basis function methods

Abstract

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.

Introduction

In [1], the adaptive RBF methods have been developed for solving ODEs. Various classical ordinary differential equation (ODE) solvers such as the Euler’s method, the midpoint method (also called the leapfrog method) and the Adams methods have been modified by replacing the polynomial basis with the MQ-RBFs. The key idea of the adaptive RBF methods for ODEs is that the RBF methods are defined with a free parameter (also known as a shape parameter) that can be adjusted depending on the local conditions of the solution. The classical ODE solvers such as finite difference methods based on the polynomial interpolation, however, have no room to improve the local accuracy and convergence unless the order of the numerical method is increased. That is, the polynomial method yields only a fixed order of convergence once the order of the method is fixed. For this case, one may not be able to take advantage of good properties of the solution, e.g. when the local derivatives of the solution are available.

But if we substitute RBFs with the free parameter for the polynomial basis, we can exploit the free parameter so that the free parameter is determined based on the local smoothness of the solution, i.e., derivative(s) in order to terminate the leading truncation error term(s). Similar idea was also applied to the ENO and WENO schemes for solving hyperbolic PDEs in [2], [3] and to error inhibiting scheme in [4].

In [1], it was shown that the adaptive MQ-RBF methods enhance the order of accuracy of the classical methods. The order enhancement was achieved by terminating the leading truncation error term(s). For the construction of the adaptive method, the truncation errors are first found as a Taylor series in terms of the grid spacing. If RBFs are used for the classical ODE solvers, each term of the truncation errors involves the shape parameter in its coefficient. The adaptive method chooses the optimal value of the shape parameter so that the coefficient of the truncation error term vanishes. The vanishing condition is given by the algebraic equation in terms of the shape parameter. The optimal value of the shape parameter is obtained by solving the given algebraic equation. In principle, the arbitrary number of the truncation error terms can be removed as long as there exists the solution of the algebraic equation. Thus, not only the first leading error term but also multiple error terms can be terminated, enhancing accuracy by several orders. That is, there exists a sequence of the shape parameters for different orders of accuracy for the given form of the method and the given number of grid points. This results in a convergence in terms of the shape parameter to the arbitrary order as far as the algebraic equations have solutions.

There have been various studies on finding the optimal values of the shape parameter for the RBF approximation for accuracy or stability [5], [6], [7], [8]. Our research is different from those in that, in our research, the optimal shape parameter is given in an exact form by terminating the truncation errors and then the exact form is approximated by the given information of the local solution. The modified methods for ODEs with the MQ-RBFs were proven to maintain enhanced accuracy and order of convergence. Furthermore, the adaptive MQ-RBF methods yield much larger stability regions than the classical finite difference solvers. As such, the proposed method has advantages over the existing methods. For example, the finite difference method based on polynomials that utilizes the adaptivity of the grid spacing can enhance accuracy but yields the fixed order of convergence yet while the proposed method provides a variable convergence. If the grid spacing adaptivity is combined with the shape parameter adaptivity it may result in a much enhanced method.

Gaussian RBFs are most popular RBFs used in RBF research due to the fact that they are similar to polynomials and easy to handle. Thus we need to develop the adaptive RBF methods based on the Gaussian RBFs along with the MQ-RBFs for the completion of the theory. Furthermore we need to compare those MQ and Gaussian RBFs for practical purposes. In order to achieve those goals, in this paper, we modify the classical ODE solvers such as the Euler method, the midpoint method (leapfrog method), the Adams–Bashforth method and the Adams–Moulton method providing the modified versions of the classical solvers and show that the adaptive Gaussian methods enhance the accuracy and convergence. We also compare the developed methods with the adaptive MQ-RBF methods. We found that the adaptive MQ-RBF method has larger stability region than the corresponding Gaussian method. Both the MQ and Gaussian RBF methods yield the desired order of convergence while one yields better accuracy than the other depending on the method chosen and the problem considered. In this research, we only consider the one-dimensional problem. The same problem in multi-dimension would be interesting and worth investigating. In [2], [3], adaptive RBF interpolations in two-dimension have been considered but not for IVPs. The adaptive RBF methods for IVPs in multi-dimension will be considered in our future research.

The paper is organized as follows. In Section 2, we provide the basic RBF interpolation with the Gaussian RBFs. In Section 3, we provide various modification of IVP solvers with the Gaussian RBF. Particularly we discuss about the consistency and stability of each modification. In Section 4, we provide numerical results. We provide the comparison with the adaptive MQ RBF methods for some IVPs.