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Rational methods for abstract linear initial boundary value problems without order reduction

Carlos Arranz-Simón, Begoña Cano, César Palencia

TL;DR

This work addresses order reduction in time integration of linear IBVPs by exploiting an $A$-stable rational approximation to $e^z$ of order $p$, yielding high-order, derivative-free time stepping for $p \le 6$. An abstract lifted IVP framework justifies the rational methods and a corresponding full discretization is analyzed, with convergence guarantees under standard stability and consistency assumptions. The authors introduce a Rational-Like (RL) method and node-sequence strategies to preserve high order while minimizing data evaluations, and validate the approach through numerical experiments on parabolic problems, showing substantial efficiency gains over conventional Runge–Kutta methods. Overall, the RL framework enables high-order, derivative-free time integration for IBVPs and delivers significant practical performance improvements in full discretizations.

Abstract

Given an $A$-stable rational approximation to $e^z$ of order $p$, numerical procedures are suggested to time integrate abstract, well-posed IBVPs, with time-dependent source term $f$ and boundary value $g$. These procedures exhibit the optimal order $p$ and can be implemented by using just one single evaluation of $f$ and $g$ per step, i.e., no evaluations of the derivatives of data are needed, and are of practical use at least for $p\le 6$. The full discretization is also studied and the theoretical results are corroborated by numerical experiments.

Rational methods for abstract linear initial boundary value problems without order reduction

TL;DR

This work addresses order reduction in time integration of linear IBVPs by exploiting an -stable rational approximation to of order , yielding high-order, derivative-free time stepping for . An abstract lifted IVP framework justifies the rational methods and a corresponding full discretization is analyzed, with convergence guarantees under standard stability and consistency assumptions. The authors introduce a Rational-Like (RL) method and node-sequence strategies to preserve high order while minimizing data evaluations, and validate the approach through numerical experiments on parabolic problems, showing substantial efficiency gains over conventional Runge–Kutta methods. Overall, the RL framework enables high-order, derivative-free time integration for IBVPs and delivers significant practical performance improvements in full discretizations.

Abstract

Given an -stable rational approximation to of order , numerical procedures are suggested to time integrate abstract, well-posed IBVPs, with time-dependent source term and boundary value . These procedures exhibit the optimal order and can be implemented by using just one single evaluation of and per step, i.e., no evaluations of the derivatives of data are needed, and are of practical use at least for . The full discretization is also studied and the theoretical results are corroborated by numerical experiments.
Paper Structure (10 sections, 5 theorems, 99 equations, 2 figures, 3 tables)

This paper contains 10 sections, 5 theorems, 99 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The IBVP
  3. The semigroup of translations
  4. The lifted IVP
  5. Rational methods for the lifted IVP
  6. The suggested method
  7. Full discretization
  8. Numerical experiments
  9. SDIRK
  10. Gauss

Key Result

Lemma 3

Let $\delta_n \in X$, $n \ge 0$, be a sequence of perturbations and let $\bar{u}_n^* \in X$, $n \ge 0$, be the solution of the perturbed recurrence with initial value $\bar{u}_0^* = \bar{u}_0 +\delta_0$. Then,

Figures (2)

  • Figure 1: Error against CPU time when integrating problem (\ref{['problem2']}) with SDIRK3 method and suggested rational SDIRK3 methods.
  • Figure 2: Error against CPU time when integrating problem (\ref{['problem2']}) with GAUSS3 method and suggested rational GAUSS3 methods.

Theorems & Definitions (15)

  • Remark 1
  • Remark 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Theorem 5
  • proof
  • Remark 6
  • Remark 7
  • ...and 5 more