Adaptive Radial Basis Function Methods for Initial Value Problems

The classical finite difference methods for solving initial value problems are based on the polynomial interpolation of the unknown solution. The expected order of convergence of every classical method is fixed regardless of the smoothness of the unknown solution. However, if the local derivatives of the solution are known or can be estimated to a certain order, such information could be used to improve the order of convergence of the local truncation error and further the order of accuracy of the global error. The aim of this paper is to utilize the radial basis function (RBF) interpolation to modify several finite difference methods and thus enhance the performance in terms of local convergence. In this work, we choose multiquadric RBFs as the interpolation basis and find the conditions of the shape parameter that could enhance accuracy. The rate of convergence of each modified method is at least the same as the original one and can be further improved by making the local truncation error vanish. In that sense, the proposed adaptive method is optimal. Compared to the linear multistep methods, the proposed adaptive RBF multistep methods exhibit higher order convergence. We provide the analysis of consistency and stability with numerical results that support our claims.