Rational methods for abstract linear, non-homogeneous problems without order reduction

TL;DR

This paper introduces stable rational methods for solving abstract linear non-homogeneous problems that achieve the desired order without reduction, requiring only one evaluation of the non-homogeneous term per step.

Contribution

It proposes new families of stable, high-order methods based on rational approximations to the exponential function, avoiding order reduction in time discretization.

Findings

Methods are of order p, matching the approximation order.

Only one evaluation of f per step is needed.

Methods effectively prevent order reduction phenomena.

Abstract

Starting from an A-stable rational approximation to $\rm{e}^z$ of order $p$, $$r(z)= 1+ z+ \cdots + z^p/ p! + O(z^{p+1}),$$ families of stable methods are proposed to time discretize abstract IVP's of the type $u'(t) = A u(t) + f(t)$. These numerical procedures turn out to be of order $p$, thus overcoming the order reduction phenomenon, and only one evaluation of $f$ per step is required.