Stabilized explicit Adams-type methods

TL;DR

This paper introduces explicit Adams-type multistep methods with extended stability intervals, providing new methods of various orders, including simple first-order methods and higher-order variants up to six, validated through numerical experiments.

Contribution

It develops explicit Adams-type methods with extended stability, including simple first-order formulas and higher-order methods up to six, with a damped variant and numerical validation.

Findings

Existence of explicit k-step methods of order one with stability interval 2k.

Simple expressions for coefficients and error constants in first-order methods.

Numerical confirmation of accuracy and stability for methods up to order six.

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 $k\geq 1$ 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.