Continuous Stage Runge-Kutta Methods and Their Structure-Preservation Properties

Abstract

Continuous stage Runge-Kutta (CSRK) methods, which were introduced around 2010, are a framework of iterative numerical methods for solving ordinary differential equations. It turned out that some CSRK methods preserve some underlying geometric structures of differential equations, such as symplecticity or energy-preservation of Hamiltonian systems. This paper reviews CSRK methods and their recent developments with emphasis on their structure-preservation properties.