New A-Stable Numerical Method for Differential Equations

Abstract

Stiff problems cause singular computational difficulties because explicit methods cannot solve these problems without rigorous limitations on the step size. To obtain high order A-stable methods, it is traditional to turn to Runge-Kutta methods or to linear multistep methods. A new multistep method is proposed for differential-algebraic equations, based in the application of estimation functions for the derivatives and the state variables, which permits the transformation of
the original system into a linear algebraic system with non-linear corrections, using the solutions of the previous steps. The originality introduced is a formula for the estimation function coefficients, which is deduced from a combined analysis of stability and convergence order. Numerical
experiments are presented comparing the new method with other classical methods.