Reid J. Gomillion | Searcharxiv

arXiv2026

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

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 stiffly implicit Runge-Kutta schemes, in particular stiffly accurate SDIRK methods. The strategy combines stage-wise clipping with a ratio-based scaling that enforces invariants and is guaranteed to yield non-negative, 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. These results show that the proposed framework provides a simple and effective way to construct positivity-preserving integrators for stiff PD systems.

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arXiv2024

Preserving Nonlinear Constraints in Variational Flow Filtering Data Assimilation

Data assimilation aims to estimate the states of a dynamical system by optimally combining sparse and noisy observations of the physical system with uncertain forecasts produced by a computational model. The states of many dynamical systems of interest obey nonlinear physical constraints, and the corresponding dynamics is confined to a certain sub-manifold of the state space. Standard data assimilation techniques applied to such systems yield posterior states lying outside the manifold, violating the physical constraints. This work focuses on particle flow filters which use stochastic differential equations to evolve state samples from a prior distribution to samples from an observation-informed posterior distribution. The variational Fokker-Planck (VFP) -- a generic particle flow filtering framework -- is extended to incorporate non-linear, equality state constraints in the analysis. To this end, two algorithmic approaches that modify the VFP stochastic differential equation are discussed: (i) VFPSTAB, to inexactly preserve constraints with the addition of a stabilizing drift term, and (ii) VFPDAE, to exactly preserve constraints by treating the VFP dynamics as a stochastic differential-algebraic equation (SDAE). Additionally, an implicit-explicit time integrator is developed to evolve the VFPDAE dynamics. The strength of the proposed approach for constraint preservation in data assimilation is demonstrated on three test problems: the double pendulum, Korteweg-de-Vries, and the incompressible Navier-Stokes equations.

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