Adaptive Adams-Type Methods Based on Algebraic, Trigonometric, and Exponential Interpolation

We consider an adaptive Adams-type multistep method for the numerical solution of ordinary differential equations based on algebraic, trigonometric, and exponential interpolation (the ATE method). For each interpolation class, the corresponding schemes are derived following the same construction as classical Adams methods. The stability properties of the trigonometric and exponential variants are analyzed, highlighting their differences relative to the algebraic method. The adaptive ATE algorithm selects one of the three schemes at each step according to a local interpolation indicator. This enables the method to adjust to different types of solution behavior. Numerical experiments are presented across a variety of differential equations, and the accuracy and execution times of the methods are compared.