A Patankar predictor-corrector approach for positivity-preserving time integration
Kamila Nurkhametova, Reid J. Gomillion, Amit N. Subrahmanya, Adrian Sandu
TL;DR
This work tackles the challenge of preserving positivity and linear invariants in stiff production-destruction systems when integrating in time. It introduces a modular Patankar predictor-corrector that post-processes base RK/SDIRK steps with stage clipping and diagonal scaling, yielding nonnegative, conservative solutions while largely maintaining the base method's order. Theoretical analysis shows the worst-case order is $\min(p, q+1)$, and extensive experiments on Robertson, MAPK, stratospheric chemistry, and KdV demonstrate robust positivity, invariant preservation, and favorable efficiency, with practical guidance on applying corrections to final versus all stages. The framework broadens the applicability of classical time integrators to PD systems and PDEs, though care is needed for certain problems (e.g., KdV) where full-stage corrections distort dynamics or explicit schemes suffer order reduction. The results justify the method as a simple, effective tool for building positivity-preserving solvers for stiff PD systems with modest overhead and wide potential applications.
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.