An explicit third-order one-step method for autonomous scalar initial value problems of first order based on adaptive quadratic Taylor approximation

TL;DR

This paper introduces a third-order explicit one-step numerical method for autonomous scalar first-order initial value problems, leveraging adaptive quadratic Taylor approximation to improve accuracy on equations like the logistic equation.

Contribution

The method uniquely uses quadratic Taylor polynomials and their exact solutions to achieve third-order accuracy in a simple explicit scheme.

Findings

Achieves third-order accuracy on autonomous scalar problems.

Error-free performance on Riccati equations like the logistic equation.

Provides an efficient alternative to traditional methods for specific problems.

Abstract

We present an explicit 1-step numerical method of third order that is error-free on autonomous scalar Riccati equations such as the logistic equation. The method replaces the differential equation by its quadratic Taylor polynomial in each step and utilizes the exact solution of that equation for the calculation of the next approximation.