A Phase/Amplification-Fitted Second-Order BDF-Type Method Using Barycentric Rational Interpolants for Orbital Initial Value Problems

Abstract We propose the design of a second-order Backward Differentiation Formula (BDF-type) method, employing barycentric rational interpolants and enhanced by a novel fitting technique for numerical solutions of second-order initial value problems (IVPs). Initially, we define barycentric rational interpolants. Subsequently, we develop a second-order BDF-type method with constant coefficients derived from these interpolants. Further, we will adjust two coefficients of the new method using the novel fitting technique. For this purpose, we express the fitting operator of the second-order BDF-type methods in the form of a theorem, which encompasses both the phase lag factor and the amplification factor. We will analyze the stability regions and local truncation error of the newly developed second-order Backward Differentiation Formula (BDF-type) method, which is based on barycentric rational interpolants. To demonstrate the superiority of this new method in terms of practical results and efficiency, we will present numerical results that include comparisons with several other methods.