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Exploring Exponential Runge-Kutta Methods: A Survey

Alessia andò, Nicolò Cangiotti, Mattia Sensi

Abstract

In this survey, we provide an in-depth investigation of exponential Runge-Kutta methods for the numerical integration of initial-value problems. These methods offer a valuable synthesis between classical Runge-Kutta methods, introduced more than a century ago, and exponential integrators, which date back to the 1960s. This manuscript presents both a historical analysis of the development of these methods up to the present day and several examples aimed at making the topic accessible to a broad audience.

Exploring Exponential Runge-Kutta Methods: A Survey

Abstract

In this survey, we provide an in-depth investigation of exponential Runge-Kutta methods for the numerical integration of initial-value problems. These methods offer a valuable synthesis between classical Runge-Kutta methods, introduced more than a century ago, and exponential integrators, which date back to the 1960s. This manuscript presents both a historical analysis of the development of these methods up to the present day and several examples aimed at making the topic accessible to a broad audience.

Paper Structure

This paper contains 18 sections, 60 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Basic definitions and construction of ExpRK methods
  3. Stiffness in a nutshell
  4. Constructing ExpRK methods
  5. Stability
  6. The historical path of ExpRK methods
  7. The first approaches
  8. The rebirth
  9. Recent advances
  10. Exponential methods of high stiff order
  11. Exponential methods for DDEs
  12. Exponential methods for mathematical modeling
  13. Examples
  14. The Six Methods
  15. Final considerations
  16. ...and 3 more sections

Figures (3)

  • Figure 1: Relative error as a function of step size, showing how accuracy of each method of order 2 (\ref{['fig:errortime_plot1']}) and order 4 (\ref{['fig:errortime_plot1_order4']}) evolves with different step choices and relative error as a function of computational time, highlighting the efficiency of methods of order 2 (\ref{['fig:errortime_plot2']}) and order 4 (\ref{['fig:errortime_plot2_order4']})
  • Figure 2: Relative error as a function of step size, showing how accuracy of each method of order 2 (\ref{['fig:error_step_plot']}) and order 4 (\ref{['fig:errortime_plot2D_1_order4']}) evolves with different step choices and relative error as a function of computational time, highlighting the efficiency of methods of order 2 (\ref{['fig:error_time_plot']}) and order 4 (\ref{['fig:errortime_plot2D_2_order4']})
  • Figure 3: Relative error as a function of step size, showing how accuracy of each method of order 2 (\ref{['fig:error_step_plotDDE_1_order2']}) and order 4 (\ref{['fig:errortime_plotDDE_1_order4']}) evolves with different step choices and relative error as a function of computational time, highlighting the efficiency of methods of order 2 (\ref{['fig:error_time_plotDDE_2_order2']}) and order 4 (\ref{['fig:error_time_plotDDE_2_order2']})

Theorems & Definitions (2)

  • Definition 2.1
  • Definition 2.2