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 Piecewise Poly-Sinc Methods for Ordinary Differential Equations

“…Theorem 3 (Estimate of Upper Bound [25]). Let y be in M α s ,β s (ϕ), analytic and bounded in D 2 , and let y (i) h (x) be the piecewise Poly-Sinc approximation in the i-th iteration.…”

“…The supremum norm has a theoretical advantage. However, its computation is slower than that of the L 2 norm [25]. Hence, we use the L 2 norm in our computations.…”

“…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.…”

“…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 2 π ≈ 0.798 for an infinite number of normal samples.…”

“…It was shown in [29] that the ratio approaches 2 π ≈ 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.…”