Inverse linear multistep methods for the numerical solution of initial value problems of ordinary differential equations

Abstract:

The well-known explicit linear multistep methods for the numerical solution of ordinary differential equations advance the numerical solution from ${x_{n + k - 1}}$ to ${x_{n + k}}$ by computing some numerical approximation from back values and then evaluating the problem defining function to obtain an approximation of the derivative. In this paper similar methods are proposed that first compute an approximation to the derivative and then compute an approximation to the exact solution, either by evaluating a suitable function, or by solving a nonlinear system of equations. The methods can be applied to initial value problems where the exact solution is explicitly given in terms of the derivative. They can also be applied in the context of the CDS technique for certain stiff initial value problems of ordinary differential equations, introduced in [1 ] and [2]. Local accuracy and stability of the methods are defined and investigated, and specific methods, containing free parameters, are given. The methods are not convergent, but they yield very good numerical results if applied to the type of problem they are designed for. Their major advantage is that they significantly reduce the amount of implicitness necessary in the numerical solution of certain problems.