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The exponential trapezoidal method for semilinear integro-differential equations

Alexander Ostermann, Nasrin Vaisi

Abstract

The exponential trapezoidal rule is proposed and analyzed for the numerical integration of semilinear integro-differential equations. Although the method is implicit, the numerical solution is easily obtained by standard fixed-point iteration, making its implementation straightforward. Second-order convergence in time is shown in an abstract Hilbert space framework under reasonable assumptions on the problem. Numerical experiments illustrate the proven order of convergence.

The exponential trapezoidal method for semilinear integro-differential equations

Abstract

The exponential trapezoidal rule is proposed and analyzed for the numerical integration of semilinear integro-differential equations. Although the method is implicit, the numerical solution is easily obtained by standard fixed-point iteration, making its implementation straightforward. Second-order convergence in time is shown in an abstract Hilbert space framework under reasonable assumptions on the problem. Numerical experiments illustrate the proven order of convergence.
Paper Structure (9 sections, 1 theorem, 53 equations, 1 figure)

This paper contains 9 sections, 1 theorem, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Setting and preliminaries
  3. Solution operator.
  4. Numerical scheme and main result
  5. The numerical method.
  6. Proof of Theorem \ref{['Theorem']}
  7. Implementation and numerical experiments
  8. Explicit representation of the solution.
  9. Numerical experiments.

Key Result

Theorem 3.2

For the solution of eq1 in the mild form variation of constant formula1, consider the exponential integrator Tra_fully_discrete. If the Assumptions A_operator, defination of f, smoothing, and assumption3b hold, then there exist constants $h_0>0$ and $C>0$ such that for all step sizes $0<h\le h_0$ an where the constant $C$ depends on $T$, but it is independent of $N$, $m$, and $h$.

Figures (1)

  • Figure 1: Temporal rate of convergence of the exponential trapezoidal method for three different problems (see text).

Theorems & Definitions (1)

  • Theorem 3.2