Euler’s Method, Taylor Series Method, Runge Kutta Methods, Multi-Step Methods and Stability. REVIEW: We start with the diﬀerential equation dy(t) dt = f (t,y(t)) (1.1) y(0) = y0 This equation can be nonlinear, or even a system of nonlinear equations (in which case y is a vector and f is a vector of n diﬀerent functions).
A family of Runge–Kutta (RK) methods designed for better stability is proposed. Authors have optimized the stability of RK method by increasing the stability region by trading some of the higher order terms in the Taylor series. signi cantly outperforms the traditional Runge-Kutta and Adams-Bashforth-Moulton methods. To quantify this, if one considers the trade-o between accuracy and computational e ort, then our augmented second-order method is approximately six times more e cient than either Runge-Kutta or Adams-