Numerical Gradient Schemes for Heat Equations Based on the Collocation Polynomial and Hermite Interpolation

TL;DR

This paper introduces a new high-order numerical gradient scheme for heat equations using collocation polynomial and Hermite interpolation, achieving high accuracy and computational efficiency.

Contribution

It presents a novel gradient scheme based on collocation polynomial and Hermite interpolation with comparable convergence order to existing methods but improved computational speed.

Findings

Achieves $O( au^2+h^4)$ convergence order under maximum norm.

Accelerates computation by using a larger space step size.

Numerical results confirm theoretical convergence and efficiency.

Abstract

As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is unconditionally stable and convergent with the order $O(\tau^2+h^4)$ under the maximum norm. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. Moreover, the convergence order of this kind of method is also $O(\tau^2+h^4)$ under the discrete maximum norm when the space step size is just twice the one of H-OCD method, which accelerates the computational process and makes the result much smoother to some extent. In addition, some corresponding analyses are made and the Richardson extrapolation technique is also considered in time direction. The results of numerical experiments are also consistent with these theoretical analysis.