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D-splitting methods: 2N -storage embedded explicit Runge-Kutta methods at any order using splitting methods

Sergio Blanes, Alejandro Escorihuela-Tomàs

Abstract

Low-storage explicit Runge-Kutta schemes are particularly popular for the numerical integration of time-dependent partial differential equations based on the method-of-lines due to their efficiency and their reduced memory requirements. We show that D-splitting methods, splitting methods on the extended phase space, can be used as high performance 2N-storage embedded explicit RK methods without a third storage register. They are pseudo-geometric methods preserving some of the qualitative properties of the exact solution up to a higher order than the order of the method. Some of their properties are analysed, to build new tailored methods, and are tested on numerical examples.

D-splitting methods: 2N -storage embedded explicit Runge-Kutta methods at any order using splitting methods

Abstract

Low-storage explicit Runge-Kutta schemes are particularly popular for the numerical integration of time-dependent partial differential equations based on the method-of-lines due to their efficiency and their reduced memory requirements. We show that D-splitting methods, splitting methods on the extended phase space, can be used as high performance 2N-storage embedded explicit RK methods without a third storage register. They are pseudo-geometric methods preserving some of the qualitative properties of the exact solution up to a higher order than the order of the method. Some of their properties are analysed, to build new tailored methods, and are tested on numerical examples.

Paper Structure

This paper contains 11 sections, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Runge-Kutta and 2$N$-storage methods
  3. D-splitting methods: Splitting methods in the duplicated phase space
  4. Working in the duplicated phase space
  5. 2$N$-storage splitting methods with a reduced number of stages
  6. Numerical examples
  7. 1D wave equation
  8. Two body Kepler problem
  9. Conclusions
  10. Acknowledgements
  11. Preservation of geometric properties: pseudo-symplectic methods

Figures (2)

  • Figure 1: Error in the solution (left) and in the mass (right) for the one-dimensional wave equation problem at final time $t_f = 50$ and $N = 128$.
  • Figure 2: Propagation of the energy error in the Kepler problem with $e = 0.8$ and $t_f = 0.8$ for three integrators, the third exhibiting pseudo-symplectic behavior. All three simulations were performed under an identical computational cost.