An explicit third-order one-step method for autonomous scalar initial value problems of first order based on quadratic Taylor approximation (Q2309709)

The authors develop an explicit third-order one-step numerical method for scalar autonomous initial value problems of ordinary differential equations (ODEs). The method is error-free for ODEs with quadratic nonlinear terms like the logistic equation and the Riccati equation. The method is based on an adaptive quadratic Taylor approximation and it is more accurate than the methods based on linearization. The convergence order of the methods is \(r+1\) for Taylor approximations of order \(r\). The step size control and issues with blow-up solutions are addressed in the numerical algorithms, which can be found as a MATLAB code in the online supplement. The quadratic Taylor method is compared with third- and fourth-order Runge-Kutta methods for some benchmark problems, the logistic equation, the Bernoulli equation, the Gompertz equation, and the flame propagation equation. The global error is often smaller by several orders of magnitude than the global error of the same order Runge-Kutta methods in the test cases.