Stability and Convergence of Multi-Product Expansion Splitting Methods with Negative Weights for Semilinear Parabolic Equations

Abstract

SIAM Journal on Numerical Analysis, Volume 64, Issue 3, Page 989-1017, June 2026.
Abstract. The operator splitting method has been widely used to solve differential equations by splitting the equation into more manageable parts. In this work, we resolve a long-standing problem—how to establish the stability of multi-product expansion (MPE) splitting methods with negative weights. The difficulty occurs because negative weights in high-order MPE methods cause the sum of the absolute values of weights to be larger than one, making standard stability proofs fail. In particular, we take the semilinear parabolic equation as a typical model and establish the stability of arbitrarily high-order MPE splitting methods with positive time steps but possibly negative weights. Rigorous convergence analysis is subsequently obtained from the stability result. Several numerical experiments validate the stability and accuracy of various high-order MPE splitting methods, highlighting their efficiency and robustness.