Vol. 16, No. 2, 2021

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Positivity-preserving adaptive Runge–Kutta methods

Stephan Nüßlein, Hendrik Ranocha and David I. Ketcheson

Vol. 16 (2021), No. 2, 155–179
Abstract

Many important differential equations model quantities whose value must remain positive or stay in some bounded interval. These bounds may not be preserved when the model is solved numerically. We propose to ensure positivity or other bounds by applying Runge–Kutta integration in which the method weights are adapted in order to enforce the bounds. The weights are chosen at each step after calculating the stage derivatives, in a way that also preserves (when possible) the order of accuracy of the method. The choice of weights is given by the solution of a linear program. We investigate different approaches to choosing the weights by considering adding further constraints. We also provide some analysis of the properties of Runge–Kutta methods with perturbed weights. Numerical examples demonstrate the effectiveness of the approach, including application to both stiff and non-stiff problems.

Keywords
positivity preserving, bound preserving, Runge–Kutta methods, linear programming
Mathematical Subject Classification
Primary: 65L06, 65L20, 65M12
Milestones
Received: 13 May 2020
Revised: 4 March 2021
Accepted: 25 April 2021
Published: 2 November 2021
Authors
Stephan Nüßlein
Department of Electrical and Computer Engineering
Technical University of Munich
80333 München
Germany
Hendrik Ranocha
Computer Electrical and Mathematical Science and Engineering (CEMSE) Division
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900
Saudi Arabia
Applied Mathematics
University of Münster
48149 Münster
Germany
David I. Ketcheson
Division of Computer, Electrical, and Mathematical Sciences and Engineering
King Abdullah University of Science and Technology (KAUST)
Thuwal 23955-6900
Saudi Arabia