Stable Positive Integral Deferred Correction Methods for Positive Dynamical Systems

In this paper, we introduce the class of Stable Positive Integral Deferred Correction (SPIDeC) methods for the numerical integration of positive dynamical systems.

The proposed framework embeds a deferred correction mechanism within an exponential-type Volterra reformulation of the underlying differential problem.

The resulting multiplicative structure guarantees the unconditional preservation of both positivity and equilibria, independently of the integration stepsize.

Arbitrarily high-order accuracy is systematically achieved through successive explicit-in-sweep corrections applied to a low-order base approximation.

From a stability viewpoint, the SPIDeC integrators are L-stable and exactly reproduce the continuous semigroup generated by diagonal linear operators.

Furthermore, when Gauss--Radau quadrature nodes are employed, the associated discrete flow asymptotically approaches a logarithmically contractive map as the number of sweeps increases, ensuring stability.

Numerical experiments are provided to validate the theoretical analysis and illustrate the practical performance of the proposed methods.