Volume Preservation by Runge-Kutta Methods

TL;DR

This paper investigates the volume-preserving properties of symplectic Runge-Kutta methods, showing they can preserve volume for broader classes of vector fields beyond Hamiltonian systems and can exactly preserve modified measures.

Contribution

It demonstrates that symplectic Runge-Kutta methods can preserve volume for a wider class of vector fields than previously known, and explores their ability to preserve modified measures.

Findings

Symplectic Runge-Kutta methods can preserve volume for non-Hamiltonian systems.

Some Runge-Kutta methods exactly preserve a modified measure.

Volume preservation is not limited to Hamiltonian systems for these methods.

Abstract

It is a classical theorem of Liouville that Hamiltonian systems preserve volume in phase space. Any symplectic Runge-Kutta method will respect this property for such systems, but it has been shown that no B-Series method can be volume preserving for all volume preserving vector fields (BIT 47 (2007) 351-378 and IMA J. Numer. Anal. 27 (2007) 381-405). In this paper we show that despite this result, symplectic Runge-Kutta methods can be volume preserving for a much larger class of vector fields than Hamiltonian systems, and discuss how some Runge-Kutta methods can preserve a modified measure exactly.