The examination of nonlinear stability and solvability of the algebraic equations for the implicit Taylor series method

Abstract

Through the availability of efficient implementations of automatic differentiation, the numerical algorithms in which (higher) derivatives of functions are used become more interesting. Such methods are, for example, the implicit Taylor series methods for solving stiff initial value problems for ordinary differential equations. By constructing relevant examples, we show that these methods (except the implicit Euler method) are not AN- or B-stable, and that for non-autonomous dissipative differential equations (except for the implicit Euler method and the implicit trapezoidal method) the algebraic equations of A-stable Taylor series methods are not solvable for all stepsizes. For autonomous dissipative equations (except for the implicit Euler method and the implicit trapezoidal method), the algebraic equations of A-stable methods do not always have a unique solution for all stepsizes.