Positivity-preserving methods for population models

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: our methods are of second order, and they are guaranteed to preserve positivity. 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.

On Adaptive Patankar Runge–Kutta methods

“…with y(t) ∈ R n and M (y(t)) ∈ R n×n . If M (y) is a graph Laplacian, then positivity and total mass are conserved by the analytical solution y(t), see [1] for details and other schemes.…”

“…4. MAPK, taken from [1], the origin is [6], consisting of six equations. It has a graph Laplacian and it is mildly stiff.…”

On Lyapunov stability of positive and conservative time integrators and application to second order modified Patankar–Runge–Kutta schemes

“…Modified Patankar-Runge-Kutta (MPRK) methods, see [7,17,18,22,23,30], guarantee positivity and conservativity of the numerical solution of positive and conservative production-destruction systems (PDS). For other recent approaches which facilitate positive and conservative numerical approximations, we refer to [1,2,5,26,29].…”