Positivity-preserving methods for ordinary differential equations

¹ Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, E-46022 Valencia, Spain
² Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB4 1LE, UK
³ Australian Research Council Centre of Excellence, for Mathematical and Statistical Frontiers (ACEMS), School of Mathematical and Physical Sciences, University of Technology Sydney, NSW 2007, Australia

^* Corresponding author: serblaza@imm.upv.es

Received: 14 October 2021
Accepted: 22 April 2022

Abstract

Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, Runge–Kutta methods and multistep methods, face an order barrier. If they preserve positivity, then they are constrained to low accuracy: they cannot be better than first order. We propose novel methods that overcome this barrier: second order methods that preserve positivity unconditionally and a third order method that preserves positivity under very mild conditions. Our methods apply to a large class of differential equations that have a special graph Laplacian structure, which we elucidate. The equations need be neither linear nor autonomous and the graph Laplacian need not be symmetric. This algebraic structure arises naturally in many important applications where positivity is required. We showcase our new methods on applications where standard high order methods fail to preserve positivity, including infectious diseases, Markov processes, master equations and chemical reactions.