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

Carlos Arranz-Simón, Cesar Palencia

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.

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

Abstract

Starting from an A-stable rational approximation to of order , families of stable methods are proposed to time discretize abstract IVP's of the type . These numerical procedures turn out to be of order , thus overcoming the order reduction phenomenon, and only one evaluation of per step is required.
Paper Structure (7 sections, 5 theorems, 87 equations, 4 tables)

This paper contains 7 sections, 5 theorems, 87 equations, 4 tables.

Table of Contents

  1. 2000 Mathematical subject classifications :
  2. Introduction
  3. Preliminaries and notation
  4. Order reduction for Runge-Kutta methods
  5. The method
  6. The holomorphic case
  7. Numerical illustrations

Key Result

Lemma 1

Let $X$ be a Banach space and $Y = \mathcal{C}_{ub}\left([0,\infty), X\right)$. Then the semigroup of translations $S_B(t): Y \rightarrow Y$, $t \geq 0$, defined by is a $\mathcal{C}_0$ semigroup. Its infinitesimal generator $B: D\left(B\right) \subset Y \rightarrow Y$ belongs to $\mathcal{G}\left(Y,1,0\right)$, $D\left(B\right) = \left\lbrace v \in Y / v' \in Y \right\rbrace$ and $B v = v'$ for

Theorems & Definitions (9)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof