# Adaptive Piecewise Poly-Sinc Methods for Ordinary Differential Equations
###### Abstract  
We propose a new method of adaptive piecewise approximation based on Sinc points for ordinary differential equations. The adaptive method is a piecewise collocation method which utilizes Poly-Sinc interpolation to reach a preset level of accuracy for the approximation. Our work extends the adaptive piecewise Poly-Sinc method to function approximation, for which we derived an a priori error estimate for our adaptive method and showed its exponential convergence in the number of iterations. In this work, we show the exponential convergence in the number of iterations of the a priori error estimate obtained from the piecewise collocation method, provided that a good estimate of the exact solution of the ordinary differential equation at the Sinc points exists. We use a statistical approach for partition refinement. The adaptive greedy piecewise Poly-Sinc algorithm is validated on regular and stiff ordinary differential equations.  
adaptive approximation; Poly-Sinc interpolation; Sinc methods; Lagrange interpolation; initial value problems; boundary value problems; exponential convergence; regular differential equations; stiff differential equations +
Footnote †:  
error estimate of the Poly-Sinc approximation is exponentially convergent in the number of Sinc points, provided that the exact solution belongs to the set of analytic functions.  
We propose an adaptive piecewise method, in which the points in a given partition are used as partitioning points. This piecewise property allows for a greater flexibility of constructing the polynomials of arbitrary degree in each partition. Recently, we developed an a priori error estimate for the adaptive method based on piecewise Poly-Sinc interpolation for function approximation [25]. In this work [25], we used a statistical approach for partition refinement in which we computed the fraction of a standard deviation [26; 27; 28] as the ratio of the mean absolute deviation to the sample standard deviation. It was shown in [29] that the ratio approaches \(\sqrt{\frac{2}{\pi}}\approx 0.798\) for an infinite number of normal samples. We extend the work [25] for regular and stiff ODEs. In this paper, we discuss the adaptive piecewise Poly-Sinc method for regular and stiff ODEs, and show that the exponentially convergent a priori error estimate for our adaptive method differs from that for function approximation [25] by a small constant.  
This paper is organized as follows. Section 2 provides an overview of the Poly-Sinc approximation, the residual computation, the indefinite integral approximation, and the collocation method. Section 3 discusses the piecewise collocation method, which is the cornerstone of the adaptive piecewise Poly-Sinc algorithm. In Section 4, we present the adaptive piecewise Poly-Sinc algorithm for ODEs and the statistical approach for partition refinement. We also demonstrate the exponential convergence of the a priori error estimate for our adaptive method. We validate our adaptive Poly-Sinc method on regular ODEs and ODEs whose exact solutions exhibit an interior layer, a boundary layer, and a shock layer in Section 5. Finally, we present our concluding remarks in Section 6.