A Patankar predictor-corrector approach for positivity-preserving time integration

TL;DR

This paper introduces a correction strategy for implicit Runge-Kutta methods that guarantees positivity and conservation in numerical solutions of PD systems, with minimal accuracy loss demonstrated through various numerical experiments.

Contribution

A modular correction approach for implicit Runge-Kutta schemes that enforces positivity and invariants, ensuring physically meaningful solutions in stiff PD systems.

Findings

Corrected SDIRK methods preserve positivity and invariants without significant accuracy loss.

Applying corrections only at the final stage is sufficient in practice.

For explicit schemes, the correction maintains positivity but reduces convergence to first order.

Abstract

Many natural processes, such as chemical reactions and wave dynamics, are modeled as production-destruction (PD) systems that obey positivity and linear conservation laws. Classical time integrators do not guarantee positivity and can produce negative or nonphysical numerical solutions. This paper presents a modular correction strategy that can be applied to implicit Runge-Kutta schemes, in particular SDIRK methods. The strategy combines stage-wise clipping with a ratio-based scaling that enforces invariants and is guaranteed to yield nonnegative, conservative solutions. We provide a theoretical analysis of the corrected schemes and characterize their worst-case order of accuracy relative to the underlying base method. Numerical experiments on stiff ODE systems (Robertson, MAPK, stratospheric chemistry) and a nonlinear PDE (the Korteweg-De Vries equation) demonstrate that the corrected SDIRK methods preserve positivity and invariants without significant loss of accuracy. Importantly, corrections applied only to the final stage are sufficient in practice, while applying them at all stages may distort dynamics in some cases. For explicit Runge-Kutta schemes, the correction maintained positivity but reduced convergence to first order. These results show that the proposed framework provides a simple and effective way to construct positivity-preserving integrators for stiff PD systems.