Stabilized explicit Adams-type methods

Abstract

In this work we present explicit Adams-type multistep methods with extended stability interval, which are analogous to the stabilized Chebyshev Runge--Kutta methods. It is proved that for any $k\ge 1$ there exists an explicit $k$-step Adams-type method of order one with stability interval of length $2k$. The first order methods have remarkably simple expressions for their coefficients and error constant. A damped modification of these methods is derived. In general case to construct a $k$-step method of order $p$ it is necessary to solve a constrained optimization problem in which the objective function and $p$ constraints are second degree polynomials in $k$ variables. We calculate higher-order methods up to order six numerically and perform some numerical experiments to confirm the accuracy and stability of the methods.