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2021 Positivity-preserving adaptive Runge–Kutta methods
Stephan Nüsslein, Hendrik Ranocha, David I. Ketcheson
Commun. Appl. Math. Comput. Sci. 16(2): 155-179 (2021). DOI: 10.2140/camcos.2021.16.155

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.

Citation

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Stephan Nüsslein. Hendrik Ranocha. David I. Ketcheson. "Positivity-preserving adaptive Runge–Kutta methods." Commun. Appl. Math. Comput. Sci. 16 (2) 155 - 179, 2021. https://doi.org/10.2140/camcos.2021.16.155

Information

Received: 13 May 2020; Revised: 4 March 2021; Accepted: 25 April 2021; Published: 2021
First available in Project Euclid: 29 March 2022

Digital Object Identifier: 10.2140/camcos.2021.16.155

Subjects:
Primary: 65L06 , 65L20 , 65M12

Keywords: bound preserving , linear programming , positivity preserving , Runge–Kutta methods

Rights: Copyright © 2021 Mathematical Sciences Publishers

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Vol.16 • No. 2 • 2021
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