Stability and convergence of multi-product expansion splitting methods with negative weights for semilinear parabolic equations

TL;DR

This paper proves the stability of high-order multi-product expansion splitting methods with negative weights for semilinear parabolic equations, enabling their reliable use in solving complex differential equations.

Contribution

It establishes the stability of multi-product expansion splitting methods with negative weights, a long-standing open problem, and provides convergence analysis and numerical validation.

Findings

High-order MPE splitting methods are stable with negative weights.

Numerical experiments confirm the methods' accuracy and robustness.

The methods are efficient for solving semilinear parabolic equations.

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 resolves 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 method cause the sum of the absolute values of weights 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.