Projective Integration Methods in the Runge-Kutta Framework and the Extension to Adaptivity in Time

TL;DR

This paper reformulates Projective Integration methods within the Runge-Kutta framework, extends them to include spatial and temporal adaptivity, and analyzes their stability, accuracy, and convergence both analytically and numerically.

Contribution

It demonstrates that all existing Projective Integration methods can be expressed as Runge-Kutta methods with extended tableaux and introduces adaptive schemes with proven properties.

Findings

Extended Projective Integration as Runge-Kutta methods

Development of spatially and temporally adaptive schemes

Analytical and numerical validation of stability and accuracy

Abstract

Projective Integration methods are explicit time integration schemes for stiff ODEs with large spectral gaps. In this paper, we show that all existing Projective Integration methods can be written as Runge-Kutta methods with an extended Butcher tableau including many stages. We prove consistency and order conditions of the Projective Integration methods using the Runge-Kutta framework. Spatially adaptive Projective Integration methods are included via partitioned Runge-Kutta methods. New time adaptive Projective Integration schemes are derived via embedded Runge-Kutta methods and step size variation while their accuracy, stability, convergence, and error estimators are investigated analytically and numerically.