A new approach to estimating a numerical solution in the error embedded correction framework

On the basis of the error correction method developed recently, an algorithm, so-called error embedded error correction method, is proposed for initial value problems. Two deferred equations are used to approximate the solution and the error, respectively, at each integration step. For the solution, the deferred equation, which is based on a modified Euler’s polygon including the information of both the solution and its estimated error at the previous integration step, is solved with the classical fourth-order Runge–Kutta method. For the error, the deferred equation, which is based on a local Hermite cubic polynomial with three pieces of information—the solution, its estimated error at the previous step, and the constructed solution—is solved by the seventh-order Runge–Kutta–Fehlberg method. The constructed algorithm controls the error and possesses a good behavior of error bound in a long time simulation. Numerical experiments are presented to validate the proposed algorithm.