Structure-preserving Numerical Methods for Non-local Photochemical Kinetics

In this work we present three classes of unconditionally positive numerical methods for a photochemical model governed by non-local integro-differential equations. Specifically, we design and compare dynamically-consistent approximation schemes based on non-standard finite differences discretizations, predictor-corrector approaches and direct quadrature integrators. A rigorous analysis is performed to establish the preservation of key physical properties, i.e. positivity, monotonicity and boundedness, regardless of the temporal, spatial and frequency stepsizes. Furthermore, theoretical results are provided to establish the high-order consistency and convergence of the methods. Comprehensive numerical experiments confirm the theoretical findings and allow for a detailed comparison of the performance and computational efficiency of the proposed discretizations. Applications to two case studies of interest, photoactivation of serotonin in left-right brain patterning and photodegradation of cadmium pigments in historical paintings, demonstrate the practical relevance of the proposed model and simulation techniques in addressing complex phenomena in photochemistry.