A Patankar Predictor-Corrector Approach for Positivity-Preserving Time Integration

Abstract

In many physical, biological, and chemical systems, the underlying dynamics are modeled by systems of ordinary differential equations in which state variables such as species concentrations must remain non-negative and often satisfy conservation laws. Standard time integration methods, including classical Runge-Kutta schemes, can violate these structural properties, leading to non-physical solutions. This thesis presents a novel positivity-preserving correction strategy applicable to general time integration schemes, with a particular focus on Runge-Kutta methods. The proposed method operates as a predictor-corrector framework, using algebraic post-processing to clip negative stage values and apply diagonal scaling to enforce both positivity and conservation. A series of benchmark problems, including the stratospheric reaction system, the MAPK cascade, and the Robertson reaction, is used to evaluate the performance of the corrected integrators. Results show that the corrected schemes successfully preserve qualitative properties without compromising numerical stability. Efficiency tests demonstrate that while corrections introduce overhead in some stiff regimes, they may also improve performance. Order verification experiments prove that the correction mechanism does not change the formal order. Overall, the proposed method provides a practical and effective approach to enforcing structural constraints in the numerical integration of stiff production-destruction systems.