A Counterexample to Robust Second-Order Convergence of the Strang Projector-Splitting Integrator

Abstract

The classical Strang projector-splitting integrator is widely observed to converge with order two, whereas the error analysis that remains uniform as the smallest singular value retained in the low-rank approximation tends to zero proves only order one. We show that this gap is intrinsic under the standard assumptions. We construct $3\times3$ matrix differential equations that are $C^2$ in time and smooth in the matrix variable, with rank-two initial data that satisfy uniform boundedness, Lipschitz, tangency-defect, and regularity bounds. Nevertheless, the exact-subflow Strang method has a nonzero $h^2$ term in the local error over one periodic forcing cycle consisting of four Strang steps. Repetition of that cycle rules out a global second-order bound whose constant and stepsize threshold are independent of the retained singular values. The mechanism is a rapid rotation of the factor directions associated with the small singular value: first-order consistency is preserved, but the changing projection spaces prevent the cancellation normally expected from a symmetric Strang composition. Hence the robust first-order result cannot, under these assumptions alone, be upgraded to robust second order for the classical projector-splitting method.