Modified Patankar Linear Multistep methods for production-destruction systems

TL;DR

This paper extends Modified Patankar schemes to linear multistep methods, ensuring positivity and conservation without step size restrictions, and introduces an embedding technique for high-order convergence.

Contribution

It develops a new class of linear multistep methods based on Patankar schemes that preserve positivity and invariants unconditionally, with high-order accuracy.

Findings

Methods preserve positivity for all step sizes.

Achieves arbitrarily high order of convergence.

Introduces an embedding technique for improved accuracy.

Abstract

Modified Patankar schemes are linearly implicit time integration methods designed to be unconditionally positive and conservative. In the present work we extend the Patankar-type approach to linear multistep methods and prove that the resulting discretizations retain, with no restrictions on the step size, the positivity of the solution and the linear invariant of the continuous-time system. Moreover, we provide results on arbitrarily high order of convergence and we introduce an embedding technique for the Patankar weight denominators to achieve it.