B-SERIES METHODS CANNOT BE VOLUME-PRESERVING
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Volume preservation is one of the qualitative characteristics common to many dynamical systems. However, it has been proved by Kang and Shang that e.g. Runge–Kutta (RK) methods can not preserve volume for all linear source-free ODEs (let alone nonlinear ODEs). On the other hand, certain so-called Exponential Runge–Kutta (ERK) methods do preserve volume for all linear source-free ODEs. Do such ERK methods perhaps also preserve volume for all nonlinear ODEs? Here we prove that the answer to this question is negative; B-series methods (which include RK, ERK and several more classes of methods) cannot preserve volume for all source-free ODEs. The proof is presented via the theory of K-loops, which is an extension of the theory of classical rooted trees.
Key wordsgeometric integration volume preservation B-series methods modified equations
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