On the improvement of a numerical method for solving high‐order non‐linear ordinary differential equations

Abstract

SUMMARYThere have been many approaches to solve ordinary differential equations numerically. Even though many numerical methods can provide very good approximate solutions they need considerable calculation effort, often through iterations. The advance of symbolic manipulation packages such as Maple gives the opportunity for new approaches to this type of problems. This paper will discuss an improvement to one of these new approaches enabled by the availability of these packages, to obtain a numerical solution of an initial-value problem governed by a high-order non-linear ordinary differential equation. The method we propose here is based on Zhu and Phan's earlier paper (Commun. Numer. Meth. Engng. 2003; 19:601-614) but we shall show an improved order of accuracy for the case of the highest-order derivative being of an even order. In any case it provides the ability to further increase the order of accuracy under certain conditions of differentiability. Like Zhu and Phan's (Commun. Numer. Meth. Engng. 2003; 19:601-614) original method, an important feature of the new method is that no iterations are needed except in the preparatory phase for some cases.

Improved multistep method with non‐linear corrections

“…(29) Figure 2 shows the temporal evolution of f (0) , f (1) , and f (2) , the coefficients a i and the non-linear correction term c, and the eigenvalue Ricatti of Equation (28). BDF is activated when |c| = 0.083 (between 0.4 and 1.4).…”

“…581 Figure 2. Temporal evolution of f (0) , f (1) , and f (2) , the A-coefficients, the non-linear correction term c, and the eigenvalue ricatti . Figure 4 shows the temporal evolution of y, y (1) , and y (2) .…”

“…In recent years, considerable efforts have been focused on the development of more advanced and efficient methods for stiff problems [1,2]. A potentially good numerical method for the solution of stiff systems of ordinary differential equations (ODEs) must have good accuracy and some reasonably wide region of absolute stability [3].…”

“…The general multistep method can be expressed in the form [5] k j=0 j y n− j = h k j=0 j f n− j (2) where j , j are parameters to be determined, f n = f (y n ), and y n = y(t 0 +nh), h being a constant time step. A multistep method is of order p if and only if [6] k j=0…”

“…n can be found by successive differentiation of Equation (1), that is y (1) n, j = f j (y n ) y (2) n, j = ∇ T f j (y n )y (1) n y (3) n, j = (y (1) n ) T H j (y n )y (1) n +∇ T f j (y n )y (2) n (13)…”