Modified Patankar Linear Multistep Methods for Production-Destruction Systems

• DOI: 10.1007/s10915-025-02804-5
• Publication Date: 2025-02-05T14:29:31Z
• Authors: Giuseppe Izzo; Eleonora Messina; Mario Pezzella; Antonia Vecchio
• Keywords: 65L05 (Primary) 65L06, 34B18 (Secondary); FOS: Mathematics; Patankar-type schemes · Positivity-preserving · High order · Conservativity ·Linear multistep methods; Mathematics - Numerical Analysis; Numerical Analysis (math.NA); Dynamical Systems (math.DS); Mathematics - Dynamical Systems

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 weights denominators to achieve it.