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Exponential Runge-Kutta methods for delay equations in the sun-star abstract framework

Alessia Ando', Rossana Vermiglio

Abstract

Exponential Runge-Kutta methods for semilinear ordinary differential equations can be extended to abstract differential equations, defined on Banach spaces. Thanks to the sun-star theory, both delay differential equations and renewal equations can be recast as abstract differential equations, which motivates the present work. The result is a general approach that allows us to define the methods explicitly and analyze their convergence properties in a unifying way.

Exponential Runge-Kutta methods for delay equations in the sun-star abstract framework

Abstract

Exponential Runge-Kutta methods for semilinear ordinary differential equations can be extended to abstract differential equations, defined on Banach spaces. Thanks to the sun-star theory, both delay differential equations and renewal equations can be recast as abstract differential equations, which motivates the present work. The result is a general approach that allows us to define the methods explicitly and analyze their convergence properties in a unifying way.
Paper Structure (9 sections, 6 theorems, 72 equations, 3 figures, 1 table)

This paper contains 9 sections, 6 theorems, 72 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Explicit ExpRK-methods
  3. Sun-star theory for delay equations
  4. Delay differential equations
  5. Renewal equations
  6. The abstract setting
  7. ExpRK-methods for delay equations
  8. Numerical results
  9. Conclusions

Key Result

Theorem 2.3

Let the initial value problem IVPADE satisfy assumptions A1-A2. Consider an explicit ExpRK-method that satisfies the order conditions up to order $p-1,$ as well as $\sum_{i=1}^{\nu}b_i(0)c_i^{p-1}=\frac{1}{p}$ and the conditions of order $p$ in the weak form, i.e., with $b_i(h\mathcal{A}_0)$ replace holds uniformly for $nh\in\left[0,T\right]$ for a constant $C$ that depends on $T,$ but is independ

Figures (3)

  • Figure 6.1: Error on the value at $T=2$ of the solution of \ref{['belzen']} for $\lambda=1$. Red: Euler method, compared with (dashed) straight line having slope 1. Blue: Heun method, compared with (dashed) straight line having slope 2. Orange: method \ref{['order3Butcher']}, compared with (dashed) straight line having slope 3.
  • Figure 6.2: Error on the $L^1$-norm at $T=4$ of the solution of \ref{['quadraticRE']} for $\gamma=4$ (left) and of its integrated state, defined by \ref{['j_RE']} (right). Red: Euler method, compared with (dashed) straight line having slope 1. Blue: Heun method, compared with (dashed) straight line having slope 2. Orange: method \ref{['order3Butcher']}, compared in the right plot with (dashed) straight line having slope 3.
  • Figure 6.3: Solution of \ref{['simpledaphnia']} for $r=K=\gamma=1$, $\overline{a}=3$ and $a_{\text{max}}=4$ from $t=0$ to $T=60$, computed with \ref{['order3Butcher']} and $h=10^{-2}$. Red: DDE component. Blue: RE component.

Theorems & Definitions (12)

  • Theorem 2.3
  • Remark 3.1
  • Theorem 3.2: diekmann95
  • Theorem 3.3: dgg07
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • proof
  • Theorem 4.6
  • proof
  • ...and 2 more