Abstract

A family of explicit symplectic partitioned Runge-Kutta methods are derived with effective order 3 for the numerical integration of separable Hamiltonian systems. The proposed explicit methods are more efficient than existing symplectic implicit Runge-Kutta methods. A selection of numerical experiments on separable Hamiltonian system confirming the efficiency of the approach is also provided with good energy conservation.

Highlights

  • Over the last few decades, a lot of progress has been made in developing numerical methods for ordinary differential equations (ODEs), which can produce efficient, reliable and qualitatively correct numerical solutions by preserving some qualitative features of the exact solutions [1,2]

  • We have extended the idea of Butcher to construct symplectic effective order partitioned Runge–Kutta (PRK) methods which are explicit in nature and are less costly than symplectic implicit RK methods

  • PRK methods, we construct two main PRK methods, two starting and two finishing PRK methods such that the two starting methods are applied once at the beginning followed by n iterations of the main PRK methods and the two finishing methods are used at the end

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Summary

Introduction

Over the last few decades, a lot of progress has been made in developing numerical methods for ordinary differential equations (ODEs), which can produce efficient, reliable and qualitatively correct numerical solutions by preserving some qualitative features of the exact solutions [1,2]. The second important property is that the phase flow is symplectic which imply that the motion along the phase curve retains the area of a bounded sub-domain in the phase space We need such numerical methods which can mimic both properties of the Hamiltonian systems. Butcher [5] tried to overcome this complexity of order barrier by presenting the idea of effective order He implemented his idea on RK method of order 5 and was able to construct explicit RK methods of effective order 5 with just 5 internal stages [5]. Butcher used the effective order technique on symplectic RK methods for the numerical integration of Hamiltonian systems [10,11]. We have extended the idea of Butcher to construct symplectic effective order PRK methods which are explicit in nature and are less costly than symplectic implicit RK methods. PRK methods, we construct two main PRK methods, two starting and two finishing PRK methods such that the two starting methods are applied once at the beginning followed by n iterations of the main PRK methods and the two finishing methods are used at the end

Algebraic Structure of PRK Methods
Rooted Trees for PRK Methods and Order Conditions
Effective Order PRK Methods
Symplectic PRK Methods with Effective Order 3
Mutually Adjoint Symplectic Effective Order PRK Methods
Numerical Experiments
Kepler’s Two Body Problem
Harmonic Oscillator
Conclusions
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